MATRIX CALCULATION
Inverse matrix calculator and inverse row computin
专利名称：Inverse matrix calculator and inverse rowcomputing method发明人：有川 勇輝,坂本 健申请号：JP2018228878申请日：20181206公开号：JP2020091691A公开日：20200611专利内容由知识产权出版社提供专利附图：摘要：The purpose of this study is to provide an inverse matrix computation system capable of dealing with matrices of relatively large size even if computational resources are limited. Solution Inverse matrix calculatorA processor 102 which performs a linefundamental transformation based on the Gaussian erasure method for an enlargement matrix (a) I in an input matrix A of N rows n columns andIncludes a memory 13 for storing an n row n column matrixProcessor 102(I + 1) row (I + 1) of the extended matrix (I =0)1...For each normalization calculation of an n-1) componentAn n row n column matrix in which one row (I + 2) column component is given as one column and one columncomponent is calculated in the memory 13, and the matrix is stored in the memoryBased on the n-row n column matrix stored in memoryThe normalization calculation is repeatedfor the (I + 2) row (I + 2) column component of the calculated expansion matrix.When the calculation is completed from the first row to the n-th row, an n row n column matrix stored in the memory 13 is output as an inverse matrix A-1.Diagram申请人：日本電信電話株式会社地址：東京都千代田区大手町一丁目５番１号国籍：JP代理人：山川 茂樹,小池 勇三,山川 政樹更多信息请下载全文后查看。
MATRIX CALCULATION
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错误提示信息说明 singular 奇异阵 Badly scaled 病态阵 inf Matlab用来表示无穷大的专用 变量，可以避免零溢出（zero overflow） Inner matrix dimensions must agree 矩阵相乘出错：维数不对
c=a' c=a.'
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三、关系运算和逻辑运算
MATLAB对此类运算符有如下规定： 所有的关系表达式或逻辑表达式中，任何非0 数都是“逻辑真”，只有0才是“逻辑假” 关系表达式或逻辑表达式的计算结果是一个 由0和1组成的“逻辑数组(Logical Array)”,数组 中1表示真，0表示假 逻辑数组是一种特殊的数值数组，对“数值 数组”操作有关命令和函数也适用它；同时又 具有自身的特殊用途 优先级别： 算术运算关系运算逻辑运算
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7、矩阵的转置
a' 或 a. '
a': 矩阵a的共轭转置
a. ': 矩阵a的转置
若a是实数矩阵，则a'==a. '
【例2-14】矩阵a为[1 2 3;4 5 6;7 8 9],计算a的转置
a=[1 2 3;4 5 6;7 8 9]; c=a'
【例2-15】矩阵a为[1+2i 3+4i],计算a的转置。 a=[1+2i 3+4i];
数组B与标量c间的乘除运算
c.*B ==B.*c: B的每一个元素乘以c c./B==B.\c : c除以 B的每一个元素 c.\B ==B./c: B的每一个元素除以c
电力系统专业英语词汇
Aadmittance导纳shunt admittance并联导纳angle角度，角internal angle ( of an alternator)(交流发电机)内角Bbus母线balancing bus平衡母线infinite bus无穷大母线load bus负荷母线passive bus无源母线slack bus松驰结点voltage controlled bus电压控制母线busbar节点，母线reference busbar参考节点Ccalculation计算load flow calculation负荷潮流计算network calculation网络计算power-flow calculation潮流计算short-circuit calculation短路计算capability能力short-circuit current withstandcapability短路电流耐受能力capacity容量available capacity可用容量capacity of synchronous condenser 调相容量installed capacity装机容量transmission capacity of a tie(line)联络线的输送容量center中心oscillating center振荡中心characteristic特性，特征power-frequency characteristic功率-频率调节特性steady-state load characteristic 稳态负荷特性transient load characteristic暂态负荷特性closing合闸compensation补偿，校正over compensation过补偿reactive power compensation无功功率补偿series compensation串联补偿shunt compensation并联补偿under compensation欠补偿control控制controlling power range控制功率范围power frequency control功率频率控制voltage control电压控制continuity连续性，持续性continuity criteria连续性指标continuity of supply供电连续性conversion变换network conversion网络变换star-polygon conversion星形-多角形变换star-triangle conversion星形-三角形变换cost价格，费用cost of attendance运行费用cost of construction制造成本cost of electric energy电能成本cost of fuel燃料费用cost of maintenance维护费cost of operation运行费cost of power production电力生产成本cost of power station电站造价cost of upkeep维修费cost of per kilowatt每千瓦造价，单位造价cost per kWh lost每度电停电损失outage cost 停电费用current电流current in the fault故障点电流fault current故障电流short-circuit current(through anelement) (通过元件的)短路电流surge current冲击电流curve曲线load curve负荷曲线load duration curve负荷持续时间曲线s tability curve稳定特性曲线swing curve振荡曲线Ddeviation偏差，偏移deviation of synchronous time同步时间偏差frequency deviation频率偏差voltage deviation电压偏差diagram图，图表topological diagram of a network网络拓扑图droop下垂，衰减droop of a system系统静特性droop of a unit机组的静特性duration（持续）时间，工作时间(annual) equivalent interruption duration (年度)等效停电时间(monthly) equivalent interruption duration (月度)等效停电时间switching duration切换时间Eenergy能，电量energy shortfall电量短缺regulation energy of a system系统的调节能量estimation估计state estimation状态估计Ffactor因数，系数power factor 功率因数redundancy factor冗余系数fault故障，错误fault analysis故障分析fault clearance故障切除fault clearing time故障时间fault location故障位置forecast预测forecast of electric demand电力需求预测load forecast负荷预测frequency频率，周波frequency drift频率漂移frequency reduction频率下降reference frequency基准频率Ggeneration发电，发生annual generating energy年发电量gross generation总发电量grid电网，网，高压输电网transmission grid输电网Iimpedance阻抗fault impedance故障阻抗longitudinal impedance纵向阻抗series impedance串联阻抗surge impedance波阻抗interconnection (of a power system)电力系统的互联Llink联接，环节asynchronous link异步联络线link in a system系统联络线limit极限，极限值，范围limit capacity极限容量limit voltage极限电压limit of power功率极限，功率范围limit of self-extinguishingcurrent自熄弧电流的极限值load负荷(载)controllable load可控负荷full load满负荷(载)generation load发电负荷gross load总负荷interruptable load可断负荷load center负荷中心load density负荷密度load flow潮流load in a system系统负荷load recovery负荷恢复load reduction 减负荷load shedding卸负荷load transfer负荷转移loss of load失负荷natural load of a line线路自然负荷non-load空载负荷peak load 尖峰负荷transferable load 可调负荷system load control 系统的负荷控制loss 损耗active loss 有功损耗active-power loss 有功(功率)损耗allowable voltage loss容许电压损耗reactive-power loss 无功(功率)损耗Mmanagement管理generation forecast management发电预测管理matrix 矩阵bus admittance matrix母线导纳矩阵bus impedance matrix母线阻抗矩阵incidence matrix 关联矩阵mesh impedance matrix网络阻抗矩阵sparse matrix 稀疏矩阵Y bus matrix Y母线矩阵Z bus matrix Z母线矩阵meter（测量仪）表，（测量）计flicker meter 闪变仪mode方式maximum operational mode最大运行方式minimum operational mode最小运行方式normal operational mode正常运行方式Nnetwork网active equivalent network有源等效网络electrical power network电力网equivalent network等效网络network splitting电网解列passive equivalent network无源等效网络Ooperation运行asymmetric operation不对称运行asynchronous operation of asynchronous machine 同步电机的异步运行incomplete phase operation 非全相运行interconnected operation 互联运行isolated operation 孤立运行out of step operation失步运行parallel operation并联运行radial operation 幅射运行ring operation环式运行separate network operation电网分列运行unwanted operation异常运行synchronous operation of a system 系统的同步运行synchronous operation of machine 电机的同步运行oscillations振荡subsynchronous oscillations次同步振荡overload过负荷Pplan 计划，设计，方案planning of power network 电网规划planning of power sources电源规划power system planning电力系统规划power 功率active power 有功hydro waiting power 水电空闲容量obstructed power受阻容量power demand from a system 系统的功率需量power flow潮流, 电力潮流power-flow distribution 潮流分布power line 电力线power shortfall功率短缺rated power 额定功率reactive power 无功reference power 基准功率reserve power 备用容量reserve power of a system 系统的备用容量short-circuit power短路容量working power工作容量Qquality质量quality of supply电能质量Rreactance 电抗system reactance系统电抗reclosure重合闸successful reclosure 重合闸成功unsuccessful reclosure重合闸不成功reserve备用spinnig reserve旋转备用static reserve静止备用resonance 谐振hyposynchronous resonance次同步谐振subsynchronous resonance次同步谐振ring环，圈ring closing闭环ring opening开环Sschedule计划generation schedule 发电计划stability稳定conditional stability有条件的稳定frequency stability 频率稳定性inherent stability固有稳定load stability负荷稳定性power system stability 电力系统稳定stability diagram稳定图stability disruption 稳定破坏stability limit 稳定极限stability margin稳定裕度stability range稳定范围stability zone稳定区steady state stability 静态稳定transient stability 暂态稳定voltage stability 电压稳定性supply供给，电源duplicate supply双电源single supply单电源stand-by supply备用电源surplus or deficit of thermal power 火电盈亏system 系统electrical power system电力系统interconnected systems 互联诸系统isolated system孤立系统power system电力系统mesh of a system单网系统meshed system多网系统overhead system架空系统radial system放射系统system configuration 系统结构图system constants系统常数system pattern 系统模式system parameters系统参数tree system链式系统underground system地下系统Ttransformer变压器main transformer主变压器transmission传输，输送transmission line输电线路transmission network输电网transmission system 输电系统transmission voltage 输电电压transmission wire 输电线tripping跳闸final tripping最终跳闸Vvoltage电压equivalent voltage flicker等值电压闪变equivalent voltage fluctuation等值电压波动highest voltage of a system系统的最高电压line voltage drop线路电压降lowest voltage of a system系统的最低电压nominal voltage of a system 系统的标称电压operating voltage in a system系统的运行电压reference voltage 基准电压voltage dips 电压骤降voltage flicker 电压闪变voltage fluctuation电压波动voltage recovery 电压恢复B备用reserve备用电源stand-by supply备用容量reserve power闭环ring closing变压器transformer变换conversion并联运行parallel operation并联导纳shunt admittance并联补偿shunt compensation波阻抗surge impedance不对称运行asymmetric operation补偿compensationC参考节点reference busbar潮流load flow潮流power flow潮流计算power-flow calculation潮流分布power-flow distribution重合闸reclosure重合闸成功successful reclosure重合闸不成功unsuccessful reclosure冲击电流surge current串联阻抗series impedance串联补偿series compensation次同步谐振hyposynchronous resonance 次同步谐振subsynchronous resonance 次同步振荡subsynchronous oscillationsD单电源single supply单网系统mesh of a system导纳admittance等效网络equivalent network等值电压波动equivalent voltagefluctuation等值电压闪变equivalent voltage flicker (年度)等效停电时间(annual) equivalent interruption duration(月度)等效停电时间(monthly)equivalent interruption duration 电机的同步运行synchronous operation of machine电抗reactance电量energy电量短缺energy shortfall电流current电能成本cost of electric energy电能质量quality of supply电力网electrical power network电力线power line电力系统power system电力系统electrical power system电力系统的互联interconnection (of a power system)电力系统规划power system planning电力系统稳定power system stability电力需求预测forecast of electric demand 电力生产成本cost of power production 电压voltage电压波动voltage fluctuation电压闪变voltage flicker电压控制voltage control电压控制母线voltage controlled bus电压骤降voltage dips电压稳定性voltage stability电压恢复voltage recovery电压偏差voltage deviation电源supply电源规划planning of power sources电网grid电网规划planning of power network电网解列network splitting电网分列运行separate network operation 电站造价cost of power station多网系统meshed system短路容量short-circuit power短路计算short-circuit calculation(通过元件的)短路电流short-circuit current(through an element)短路电流耐受能力short-circuit current withstand capability地下系统underground systemE额定功率rated powerF发电generation发电负荷generation load发电计划generation schedule发电预测管理generation forecast management范围limit方案plan方式mode放射系统radial system费用cost非全相运行incomplete phase operation 负荷(载) load负荷潮流计算load flow calculation负荷持续时间曲线load duration curve 负荷恢复load recovery负荷密度load density负荷母线load bus负荷曲线load curve负荷中心load center负荷转移load transfer负荷稳定性load stability负荷预测load forecast幅射运行radial operationG功率power功率因数power factor功率极限limit of power功率范围limit of power功率短缺power shortfall功率-频率调节特性power-frequency characteristic功率频率控制power frequency control 工作容量working power供电连续性continuity of supply过补偿over compensation过负荷overload故障fault 故障分析fault analysis故障切除fault clearance故障时间fault clearing time故障位置fault location故障电流fault current故障点电流current in the fault故障阻抗fault impedance孤立系统isolated system孤立运行isolated operation固有稳定inherent stability估计estimation管理management关联矩阵incidence matrixH合闸closing火电盈亏surplus or deficit of thermal power互联诸系统interconnected systems 互联运行interconnected operation环式运行ring operation环，圈ringJ计划plan计划schedule计算calculation基准电压reference voltage基准频率reference frequency基准功率reference power极限limit极限值limit极限容量limit capacity极限电压limit voltage机组的静特性droop of a unit架空系统overhead system价格cost尖峰负荷peak load减负荷load reduction角度angle节点busbar静止备用static reserve静态稳定steady state stability矩阵matrixK开环ring opening可断负荷interruptable load可控负荷controllable load可调负荷transferable load可用容量available capacity控制control控制功率范围controlling power range空载负荷non-loadL联接link联络线的输送容量transmission capacity of a tie (line)链式系统tree system连续性continuity连续性指标continuity criteriaM满负荷(载) full load每千瓦造价，单位造价cost of per kilowatt 每度电停电损失cost per kWh lost母线bus母线busbar母线导纳矩阵bus admittance matrix母线阻抗矩阵bus impedance matrixN能energy能力capability(交流发电机)内角internal angle ( of an alternator)年发电量annual generating energyP偏差deviation偏移deviation频率frequency频率偏差frequency deviation频率漂移frequency drift频率下降frequency reduction 频率稳定性frequency stability平衡母线balancing busQ欠补偿under compensation切换时间switching duration曲线curveR燃料费用cost of fuel容量capacity冗余系数redundancy factor容许电压损耗allowable voltage lossS闪变仪flicker meter输送transmission输电线路transmission line输电网transmission network输电网transmission grid输电线transmission wire输电系统transmission system输电电压transmission voltage失步运行out of step operation失负荷loss of load双电源duplicate supply受阻容量obstructed power水电空闲容量hydro waiting power松驰结点slack bus损耗lossT特性characteristic特征characteristic同步电机的异步运行asynchronous operation of a synchronous machine 同步时间偏差deviation of synchronous time跳闸tripping调相容量capacity of synchronous condenser停电费用outage cost图，图表diagramW网network网络变换network conversion网络计算network calculation网络拓扑图topological diagram of a network网络阻抗矩阵mesh impedance matrix稳定stability稳定范围stability range稳定图stability diagram稳定区stability zone稳定破坏stability disruption稳定极限stability limit稳定裕度stability margin稳定特性曲线stability curve稳态负荷特性steady-state loadcharacteristic维护费cost of maintenance维修费cost of upkeep无功reactive power无功(功率)损耗reactive-power loss无功功率补偿reactive powercompensation无源等效网络passive equivalent network 无源母线passive bus无穷大母线infinite busX系数factor系统system系统电抗system reactance系统参数system parameters系统常数system constants系统模式system pattern系统结构图system configuration系统联络线link in a system系统静特性droop of a system系统负荷load in a system系统的负荷控制system load control系统的同步运行synchronous operation ofa system系统的运行电压operating voltage in asystem系统的最高电压highest voltage of a system系统的最低电压lowest voltage of a system系统的标称电压nominal voltage of a system系统的备用容量reserve power of a system系统的调节能量regulation energy of a system系统的功率需量power demand from a system下垂，衰减droop稀疏矩阵sparse matrix线路电压降line voltage drop线路自然负荷natural load of a line谐振resonance卸负荷load shedding星形-多角形变换star-polygon conversion 星形-三角形变换star-triangle conversion 旋转备用spinnig reserveYY母线矩阵Y bus matrix异常运行unwanted operation异步联络线asynchronous link因数factor有功active power有功损耗active loss有功(功率)损耗active-power loss有条件的稳定conditional stability有源等效网络active equivalent network预测forecast运行operation运行费cost of operation运行费用cost of attendanceZZ母线矩阵Z bus matrix暂态负荷特性transient load characteristic 暂态稳定transient stability振荡oscillations振荡中心oscillating center振荡曲线swing curve正常运行方式normal operational mode周波frequency中心center主变压器main transformer质量quality制造成本cost of construction状态估计state estimation装机容量installed capacity总发电量gross generation总负荷gross load最终跳闸final tripping阻抗impedance纵向阻抗longitudinal impedance自熄弧电流的极限值limit of self- extinguishing current最大运行方式maximum operational mode 最小运行方式minimum operational mode。
数学专业英语词汇(M)
数学专业英语词汇(M)数学专业英语词汇(M)数学专业英语词汇(M)mach angle 马赫角mach cone 马赫锥mach number 马赫数machine computation 机破算machine computing 机破算machine equation 机平程machine language 机骑言machine word 计算机语mackey topology 麦基拓扑maclaurin expansion 马克劳林展开maclaurin formula 马克劳林公式macro instruction 广义指令macrooperation 大运算macroparameter 宏观参数macrostatistics 宏观统计学magic circle 幻圆magic cube 幻立方magic figure 幻图magic square 纵横图magnetic head 磁头magnetic store 磁存储器magnetic tape 磁带magnetohydrodynamics 磁铃力学magnitude 量main diagonal 衷角线main program 痔序major axis 长轴major cycle 大循环major premise 大前提major term 大词majorant 强级数majorant criterion 比较检验majorant series 强级数majority 多数majority decision function 多数判定函数majority function 强函数majority game 强对策majorized sequence 优化序列majorized series 优化级数mal posed problem 不适定问题malfunction 错误动作maltiple classification 廖分类manifold 廖manifold classification 廖分类manifold of flags 旗廖manifold without boundary 无边廖manipulation 操作mannheim curve 曼海姆曲线mantissa 尾数many body problem 多体问题many dimentional sepce 多维空间many valued composition law 多值合成律many valued function 多值函数many valued logic 多值逻辑many valued mapping 多值映射map 映射map coloring problem 地图着色问题map projection 地图投影mapping 映射mapping cone 映射锥mapping cylinder 映射柱mapping function 映射函数mapping norm 映射范数mapping of sets 集映射mapping of the boundary 边缘映射mapping space 映射空间mapping theorem 映射定理mapping transformation 映射变换marginal density 边缘密度marginal distribution 边缘分布marginal distribution density function 边缘分布密度函数marginal distribution function 边缘分布函数mark 记号market model 市场模型marking function 标记函数markoff chain 马尔可夫链markov chain 马尔可夫链markov decision process 马尔可夫决策过程markov matrix 马尔可夫矩阵markov process 马尔可夫过程markov transform 马尔可夫变换marriage problem 配对问题mass 质量master program 痔序master sample 标准样本matching 匹配matching theorems 匹配定理material implication 实质蕴涵mathematical 数学的mathematical analysis 数学分析mathematical approximation 数学近似法mathematical constant 数学常数mathematical expectation 期望值mathematical formula 数学公式mathematical induction 数学归纳法mathematical logic 数理逻辑mathematical model 数学模型mathematical pendulum 数学摆mathematical physics 数学物理mathematical programming 数学规划mathematical random sample 数学随机样本mathematical statistics 数理统计mathematics 数学mathieu equation 马提厄方程mathieu function 马提厄函数mathieu group 马提厄群matricial rank 矩阵的秩matrix 矩阵matrix algebra 矩阵代数matrix analysis 矩阵分析matrix calculation 矩阵计算matrix element 矩阵元matrix equation 矩阵方程matrix factorization method 矩阵因子分解方法matrix form 矩阵形式matrix function 矩阵函数matrix game 矩阵对策matrix group 矩阵群matrix inversion 矩阵求逆matrix norm 矩阵范数matrix of coefficients 系数矩阵matrix of the transformation 变换矩阵matrix operator 矩阵算子matrix power series 矩阵幂级数matrix product 矩阵积matrix representation 阵表示matrix ring 矩阵环matrix semigroup 矩阵半群matrix series 矩阵级数matrix solution 矩阵解matrix transformation 矩阵变换matrix tree theorem 矩阵狮理matrix unit 矩阵单位matroid 矩阵胚maximal abelian extension 最大阿贝耳扩张maximal chain 连通链maximal element 极大元maximal equivalent orber 极大整环maximal hermitian operator 最大埃尔米特算子maximal ideal 极大理想maximal ideal space 极大理想空间maximal operator 最大算子maximal order 极大整环maximal principle 最大值原理maximal separable extension 极大可分扩张maximal strip 极大带maximal tree 最大树生成树maximality 极大性maximin 极大极小maximization 极大化maximizing sequence 极大化序列maximum 最大maximum condition 极大条件maximum deviation 最大偏差maximum ergodic theorem 极大遍历定理maximum likelihood equations 极大似然方程maximum likelihood estimating function 极大似然估计量maximum likelihood estimator 极大似然估计量maximum likelihood method 极大似然法maximum likelihood principle 极大似然法maximum matching 极大匹配maximum modulus principle 最大模原理maximum number 最大数maximum of a function 函数最大maximum or minimum condition 极大或极小条件maximum point 最大点maximum principle 最大值原理maximum problem 极大值问题maximum solution 最大解maximum term 极大项maximum value 绝对极大值maxwell boltzmann distribution law 麦克斯韦玻耳兹曼分布律maxwell's distributlon law 麦克斯事分布律maxwell's equations 麦克斯事方程meager set 贫集mean 平均mean continuity 中数连续性mean convergence 平均收敛mean convergence of p th order p阶平均收敛mean curvature 平均曲率mean curvature of surface 曲面的平均曲率mean density 平均密度mean derivative 平均微商mean deviation 平均偏差mean difference 平均差mean error 平均误差mean life 平均寿命mean number 平均数mean ordinate 平均纵坐标mean pay off 平均支付mean proportional 比例中项mean square 均方mean square contingency 均方列联mean square deviation 方差mean square of error 误差的均方mean square value 均方值mean term 内项mean type 平均型mean value 平均值mean value method 平均值法mean value theorem 平均值定理mean vector 均值向量measurability 可测性measurable 可测的measurable function 可测函数measurable mapping 可测映射measurable set 可测集measure 测度measure of dispersion 离差的度量measure of skewness 偏度measure preserving transformation 保测变换measure space 测度空间measure theory 测度论measure zero 零测度measurement 测量measuring error 测量误差measuring rule 量尺mechanics 力学mechanism 机构median 中位数median line 中线median point 中点mediant 中间数medium 媒体meet 交meet homomorphic image 保交同态像meet irreducible element 交不可约元素mega 兆member 项member of an equation 方程的端边memory 存储器memory capacity 存储容量memory cell 存储单元memory register 存储寄存器mental arithmetic 心算meridian 子午线meromorphic differential 亚纯微分meromorphic function 亚纯函数meromorphic function element 亚纯函数元素meromorphic mapping 亚纯映射meromorphism 亚纯映射meromorphy 亚纯mesh point 网格点mesh size 网格大小mesokurtic distribution 常峰态分布meta axiom of choice 亚选择公理metabelian group 亚交换群metacompact space 亚紧空间metaharmonic function 亚低函数metalanguage 元语言metalogic of predicates 谓词元逻辑metatheorem 元定理meter 米method of approximation 近似法method of artificial variables 人工变量法method of balayage 扫除法method of characteristic curves 特者法method of comparison 比较法method of conjugate gradients 共轭梯度法method of difference 差分法method of elimination 消元法method of estimation 估计法method of exhaustion 穷竭法method of false position 试位法method of finite elements 有限元法method of fractional steps 分步法method of integration of partial differential equations 偏微分方程的积分法method of iteration 迭代法method of partial fractions 部分分数法method of perturbation 扰动法method of potentials 起脚石法method of power series 幂级数法method of principal axes 轴法method of principal components 种量法method of regularization 正则化法method of residues 剩余法method of runge kutta type 朗格库塔型的方法method of steepest ascent 最速上升法method of steepest descent 最速下降法method of successive approximation 逐次近似法method of undetermined coefficients 比较法metre 米metric 度量metric coefficient 度量系数metric connection 度量联络metric form 度量形式metric normal form of quadratic form 二次形式的度量标准形式metric space 度量空间metric subspace 度量子空间metric tensor 基本张量metric topology 度量拓扑metrically convex subset 度量凸子集metrically dense 度量的稠密metrizability 可度量性metrizable 可度量化的metrizable group 可度量化群metrizable uniform space 可度量化一致空间metrization 度量化metrization theorem of urysohn 乌里申度量化定理microlocal analysis 微局部分析mid square method 平方取中法middle term 中项midperpendicular 中垂线midpoint 中点midrange 中列数millimeter 毫米million 百万minimal automaton 极小自动机minimal basis 极小基minimal disjunctive normal form 极小析取范式minimal element 极小元素minimal generating set 不可约生成集minimal graph 极小图形minimal manifold 极小簇minimal model 极小模型minimal polynomial 极小多项式minimal propositional calculus 极小命题演算minimal solution 极小解minimal submanifold 极小子廖minimal sufficient estimator 最小充分估计量minimal sufficient statistic 最小充分统计量minimal surface 极小曲面minimal type 极小类型minimal variety 极小簇minimality 极小性minimax 极小极大minimax decision function 极小极大判决函数minimax inequality 极小极大不等式minimax principle 极小极大原理minimax solution 极小极大解minimax strategy 极小极大策略minimax theorem 极小极大定理minimization 极小化minimizing method 极小化法minimizing sequence 极小化序列minimum 最小minimum condition 极小条件minimum covering 极小覆盖minimum density 极小密度minimum integral 极小解minimum modulus 最小模minimum modulus principle 最小模原理minimum number 最小数minimum of a function 函数的最小minimum point 极小点minimum principle 极小原理minimum problem 极小问题minimum solution 极小解minimum value 最小值minimum with a condition 条件极小minimum with a constraint 条件极小minkowski approximation theorem 闵可夫斯基逼近定理minkowski inequalities 闵可夫斯基不等式minkowskian addition 闵可夫斯基加法minkowskian linear combination 闵可夫斯基线性组合minkowskian space 闵可夫斯基空间minor 子式minor arc 劣弧minor axis 短轴minor cycle 小循环minor determinant 子行列式minor premise 小前提minor term 小词minorant 弱函数minorant function 弱函数minuend 被减数minus infinity 负无穷大minus mark 负号minute 分miscalculation 计算误差;计算误差missing plot technique 缺区补救技术missing value 缺少值mistake 错误mix 混合mixed area 混合面积mixed concomitant 混合相伴式mixed differential parameter 混合微分参数mixed distribution 混合分布mixed graph 混合图形mixed group 混合群mixed ideal 混合理想mixed number 带分数mixed partial derivative 混合偏导数mixed problem 混合问题mixed side condition 混合边条件mixed strategy 混合策略mixed tensor 混合张量mixed type 混合型mixed vertex 混合顶点mixing problem 混合问题mixing ratio 混合比mixture 混合mnemonic 助记的mnemonic device 助记装置mnemonics 助记mobility 可动性modal class 众数组modal proposition 模态命题modal system 模态系统modal value 最常见的值modality logics 模态逻辑mode 众数mode of vibration 振动模式model 模型model of moving means 移动平均模型model test 模型试验model theory 模型理论modern geometry 近世几何学modification 变形modified bessel function 修正贝塞耳函数modified newton method 修正牛顿法modular 模的modular category 模范畴modular character 模特征modular equation 模方程modular figure 模图modular form 模形式modular function 模函数modular group 模群modular lattice 模格modular matrix 模矩阵modular substitution 模置换modular variety 模簇module 模module of boundaries 边界模module of homomorphisms 同态模module of program 程序的模module of quotients 商模moduli space 参模空间modulo 模modulus 绝对值;模modulus of a congruence 同余模modulus of elasticity 弹性模数modulus of periodicity 周期的模modulus of rigidity 刚性模量modus tollens 否定式moment 矩moment generating function 矩量母函数moment matrix 矩量矩阵moment of distribution 分布矩moment of force 力矩moment of inertia 惯性矩moment of momentum 动量矩moments method 矩量法momentum 动量monad 单子monge cone 蒙日锥monic 首一的monic polynomial 首一多项式monitor 监督程序monocyclic system 单循环系monodromy 单值monodromy group 单值群monodromy theorem 单值定理monogenic 单演的monogenic function 单演函数monogenic module 循环模monogyre 一次对称轴monoid 单式半群monoidal representation 单项表示monoidal transformation 单项变换monomial 单项式monomial equation 单项方程monomial factor 单项因子monomial form 单项形式monomial group 单项群monomial representation 单项表示monomorphism 单一同态monotone approximation 单灯近monotone class 单掂monotone decreasing 单递减的monotone decreasing function 单递减函数monotone function 单弹数monotone increasing function 单递增函数monotone numbering 单掂号monotone sequence 单凋列monotonic function 单弹数monotonic system of sets 单掂monotonic transformation 单典换monotonically decreasing sequence 单递减序列monotonically increasing sequence 单递增序列monotonicity 单翟monotony interval 单跌间monte carlo method 蒙特卡罗法monte carlo simulation 蒙特卡罗模拟montel space 空间moore smith convergence 穆尔史密斯收敛morphism 射morse inequalities 莫尔斯不等式morse theory 莫尔斯理论mortality rate 死亡率mortality table 死亡率表most powerful test 最大功效检定most probable duration 最可能持续时间most probable value 最常见的值most stringent test 最紧迫检验motion 运动motion equation 运动方程movable singularity 可移奇点move 步着movement 运动moving arm 移动臂moving average 移动平均moving average method 移动平均法moving frame 活动标架moving trihedral 怜三面形moving trihedron 怜三面形mrkoff process 马尔可夫过程mu continuity 连续性mu continuous function 连续函数mu integrable 可积分mu measurable 可测的mu singular 奇异multi dimensional integral 多维积分multi modal distribution 多重模态分布multi person game 多人对策multi phase sampling 多相抽样法multi purpose computer 万能计算机multi stage game 多阶段对策multi stage sampling 多级抽样法multi valued mapping 多值映射multi valuedness 多值性multiaddress 多。
基于百度地图API访问的出行时间成本矩阵计算
0引言研究背景：随着世界各国城市化进程的不断加快，城市空间的深刻变动导致设施配置与公众需求之间的矛盾逐渐突出。
1999年，中国正式迈入人口老龄化社会[1]，相比于中国，英国、日本、瑞典等发达国家更早地进入了人口老龄化社会，因此，国外在养老问题方面的研究起步更早[2]，在养老服务设施方面已经取得了一定的研究成果。
随着我国人口老龄化的不断深入，国内对养老服务设施规划与管理方面的研究也逐渐展开，然而我国养老产业仍处于起步和摸索阶段[3]。
随着老龄化程度的加深以及家庭养老功能的逐渐弱化，在中国养老服务体系中，机构养老逐渐成为一种重要的辅助养老模式[4]，其作为为老年人提供集中居住和照料服务的机构，逐渐成为国内外学者们的研究热点[5]。
研究区域：2010年8月23日，国务院批复确定郑州市为我国中部地区重要的中心城市和国家重要的综合交通枢纽[6]。
郑州市下辖6个区（二七区、中原区、回族管城区、惠济区、金水区、上街区）、1个县（中牟县）、代管5个县级市（巩义市、新密市、新郑市、登封市、荥阳市）。
河南省人民政府门户网站显示，截至2021年9月22日，郑州市总面积为7567平方千米，总人口为1260万人。
其中主城区（二七区、中原区、回族管城区、惠济区、金水区）面积约为全市总面积的1/10，却聚集了约全市1/3的人口。
郑州市主城区老年人口占比大，养老服务任务的挑战非常大。
因此，本文以郑州市主城区为研究区域，分析其养老设施的空间可达性，研究区域包括二七区、中原区、回族管城区、惠济区、金水区五个区，如图1所示。
本文根据河南省民政厅养老设施供需发布信息平台和国家统计局，并结合实地调研，共得到64条供给点（郑州市主城区养老机构）地址信息和80条需求点（郑州市主城区各街道）地址信息，数据样例见表1和表2。
———————————————————————作者简介:卢鹏羽（1996-），女，河南济源人，硕士，研究方向为管理科学。
基于百度地图API 访问的出行时间成本矩阵计算Travel Time Cost Matrix Calculation Based on Baidu Map API Access卢鹏羽LU Peng-yu（同济大学，上海201804）（Tongji University ，Shanghai 201804，China ）摘要:养老服务设施规划与管理方面的研究对于解决我国老龄化问题具有深远的意义，养老设施需求点与供给点之间时间成本矩阵的计算是很关键的一部分。
矩阵的初等变换及其应用(Elementary transformation of matrix and its application)
矩阵的初等变换及其应用（Elementary transformation of matrixand its application）Elementary transformation of matrix and its applicationWang DanElementary transformation of matrix and its applicationAbstractElementary transformation of matrix is an important method of studying matrix, and it is the core of application in linear algebra. This paper introduces some concepts and properties associated with the matrix, on the basis of matrix rank, the basis for judgment matrix is invertible, after inverse matrix equations, eigenvalues and eigenvectors, two types of standard form, and illustrate the application of elementary transformation of matrix in the above is how to play the role of.Keywords: matrix, elementary transformation, applicationThe, elementary, transformation, of, matrix, and, its, applicationsAbstractElementary transformation matrix is an important means of Matrix is the core linear algebra applications. This article briefly describes some of the concepts and propertiesassociated with the matrix as a basis, the rank of a matrix to determine whether a matrix is reversible after inverse matrix, seeking basic solutions line equations find eigenvalues, and eigenvectors, quadratic standard Shape and so on. Illustrate the elementary transformation matrix in the above applications is how to play a role.Keywords:, matrix, elementary, transformation, applicationCatalog1. introduction 62. the related concepts of matrix 72.1 definition of matrix 72.2 transpose of matrix 72.3 elementary transformation of matrix and elementary matrix 73. the application of elementary transformation of matrix 83.1, the rank of the matrix 83.2 the inverse matrix of the matrix 103.3 using elementary transformation to solve matrix equation 113.4 find the solution of linear equations 12The conditions for the existence of nonzero solutions of 3.4.1 homogeneous linear equations are 13Conditions for the existence of solutions of 3.4.2 nonhomogeneous linear equations 143.5 find the eigenvalues and eigenvectors of the matrix 153.6, use elementary transformation, two times as standard type 17Summary 19References 191. introductionIn the course of studying linear algebra, I find that the elementary transformation of matrix is very extensive and runs through the whole chapter. It is the key to solve the problem in linear algebra. Linear equations is the beginning of the elementary transformation matrix, the matrix effect can also be said to be of linear algebra, each knowledge point of linear algebra and linear algebra and matrix are closely related, each in mathematics both can play a role. Biology, economics, physics, cryptography requires knowledge of mathematics, the significance of matrix elementary transformation of matrix, as can be imagined, is the complex matrix into a simple form is easy to calculate and understand.In real life, many aspects involve the knowledge of matrices,In studying the virtual aircraft model, we will find that the operation of the matrix plays a crucial role. The plane surface appears to be smooth, but the geometric structure is perplexing, the flow equation is more difficult, must also consider other external factors, but we use the matrix knowledge to be able to solve the problem very well. There are many other applications, for example, matrix eigenvalues and eigenvectors is the key to solve many problems in physics, mechanics and engineering technology; now the game company and Bank Account confidential security, but also the use of matrix theory invented the matrix card; simulation in equipment monitoring system in engineering, radio and television; large screen display works, TV teaching, command and control center etc. mainly used matrix switcher and so on.2. concepts related to matrices2.1 definition of matrixTable is a rectangular matrix. Similar to the cross and the determinant is called a row, called vertical columns, with a line, the line and the line and the row element matrix for short note.Transpose of the 2.2 matrixLet a matrix be called a matrixFor the transpose of the matrix, rememberElementary transformation and elementary matrix of 2.3 matrices1, the following three transformations called matrices, called the matrix of the primary row (column) transform, collectively referred to as the elementary transformation of the matrix:(1) the two row (column) of the exchange matrix(2) the elements of a row (column) of a matrix are multiplied by a nonzero constant(3) a constant of the elements of a row (column) of a matrix added to the corresponding element of another row (column)Elementary row and column transformations are collectively referred to as elementary transformations2. The matrix obtained by elementary transformation of a unit matrix is called elementary matrix.Three types of elementary matrices:(1) elementary commutative matrices: the second and the second lines of the commutative unit matrix(2) the elementary multiplied matrix: the row (column) of the unit matrix takes the nonzero constant, i.e.(3) elementary doubly matrix: the first row of a unit matrix is added to the first line, or the first row is multiplied to the next columnIf the matrix is transformed into a matrix by a finite elementary transformation, it is said to be equivalent3, matrix equivalence has the following properties:(1) reflexivity, that is, the self equivalence of any matrix;(2) symmetry, that is, the equivalence of any matrix, if and equivalence;(3) transitivity is equivalent to any matrix, and if and equivalence, equivalence, and equivalence;The application of elementary transformation of 3. matrices3.1, the rank of the matrixMany methods for matrix rank, general definition method, elementary transformation method, formula method and comprehensive method, but when the specific element of the matrix is known, using elementary transformation method is for non zero row (column) number.The highest order of a nonzero divisor defined in a 3.1.1 matrix is called the rank of a matrix. That is, there is a rank order of no 0, and all orders of variables (if any) are 0, then the rank of the matrix is (or / or rank)(1)(2) the rank of the zero matrix is 0(3) the rank of a ladder matrix = the number of nonzero rows in a rowTheorem 3.1.1 the elementary transformation of a matrix does not change the rank of a matrixTheorem 3.1.2 row rank of a matrix = row rank of a matrixTheorem 3.1.3, the equivalent matrices have the same rank, but their inverse is not true, that is, the matrices with the same rank may not be equivalent, and the matrices of the same type and the same rank are equivalent to each otherFind the rank of a matrix, and give a brief introduction of the most common method:(1) definition method:If the matrix has a nonzero order, and all the sub orders (if any) are all 0, then.If there is a nonzero order in the matrix, and all of the order variables containing this order are 0.The usage of matrix rank can be calculated with simple formula omit a lot.(2) the number of zero rows in Central Africa is the rank of the matrix.This is because the elementary row transformation does not change the rank of the matrix, in addition, it can be transformed into a column ladder rank by the elementary column transformation, and the elementary transformation can be used as the standard form to obtain the rank.Example 1 find the rank of a matrix.Solution 1: take the 2 order of the upper left of the matrixHowever, there are only 3 lines in the matrix, so it is necessary to find the 3 order of the variables contained in the matrix.Solution 2: to do elementary row transformationDue to non-zero behavior 2.It can be seen that the definition method is only suitable for the calculation of simple matrix, but if it is a higher order matrix, it is very inconvenient to calculate.3.2, the inverse matrix of the matrixThe definition of 3.2.1 is set as a square matrix, if the order matrix existsHere is the rank unit matrix, which is called the invertible matrix, and is called the inverse matrix.Note (1) if it is invertible, its inverse matrix is unique, and the inverse matrix is;(2) the invertible problem of the matrix is the case of the opponent's matrix.Set the invertible matrix of order, and the inverse matrix is as follows:Example 2 is set up as a invertible square matrix, and the resulting matrix is denoted by the following line and column(1) proved to be reversible;(2) seekingProof: (1) since the left multiplication of the elementary matrix corresponds to the two rows of the interchange, so there isBecause, so the matrix is reversible(2)Example 3 uses the elementary transformation of the matrix to find the inverse matrix of the matrixSolution:soIn short, we in the inverse matrix with elementary transformation, we must first selected by elementary row transformation or elementary column transformation, note that if using elementary row transformation must be from first to last by elementary row transformation, using elementary column transformation must be from first to last by elementary column transformation.But in the inverse does not need to check whether the reversible matrix, elementary transformation can be directly obtained, if the simplest form of a square matrix transform unit is not left after the show, the original matrix is irreversible.3.3 using elementary transformation to solve matrix equation(1) if it is reversible, then(2) if it is reversible, then(3) if both are reversible, thenFirst of allAgainThis can be obtainedThe matrix equations of type can only be elementary row transformations (on the left); the pair can only be elementary column transformations (on the right)Example 4 solving matrix equationSolution: let the original equation be...therefore3.4 solving the system of linear equationsSet a system of linear equations with unknown quantitiesIts matrix form is,Among them,,,The coefficient matrix called linear equation is called the augmented matrix.Conditions for nonzero solutions of 3.4.1 homogeneous linear equations(1) the necessary and sufficient condition for the existence of nonzero solutions of homogeneous linear equations is the rank of the coefficient matrix(2) when the number of equations of a homogeneous linear equation group is less than the number of unknown quantities (m<n), there must be nonzero solutions(3) if the order matrix is square, the system of equations has nonzero solution(4) if the order matrix is square, then the system of equations has only zero solutionFirst, the coefficient matrix is transformed into a ladder matrix by using elementary row transformation, and if there is only zero solution, if there is a nonzero solution, it continues to be calculated;The ladder? Matrix to the simplest form, a non zero row non zero element corresponding to the unknown quantity, the unknown amount of free unknown quantity, revenuer, after making one of a free variable is 1, the remaining 0, basic system of solutions can be obtained.The linear combination of the solutions of the parameters is the general solution of the equationExample 5 solving linear equationsSolution: the coefficient matrix is transformed into the simplest form by elementary row transformationsoThat is, there are 2 free unknownsWith the same set of equationsFor the selection of free unknown, and transferred toThe general solution is()Represented as a vector matrixConditions for the existence of solutions of 3.4.2 nonhomogeneous linear equations(1) if the set is a matrix, then the necessary and sufficient condition for the solution of the nonhomogeneous linear equation set is that the rank of the coefficient matrix is equal to the rank of the augmented matrix(2) if the set of nonhomogeneous linear equations is solvable, thenThe solution is unique and the second set of equations has only zero solutions.(3) there are infinitely many solutions to the system of nonhomogeneous linear equations(4) the solution of a system of nonhomogeneous linear equations without elementExample 6 for solving nonhomogeneous linear equationsSolution: an elementary row transformation of the augmented matrixThat wasTherefore, the general solution of the original equation set is any constant3.5 find the eigenvalues and eigenvectors of the matrixThe definition of 3.5.1 is a matrix of order, if there exists a number and a zero dimensional column vector, theThat isSet up is called an eigenvalue of a square matrix, and nonzero column vectors are called eigenvectors of the square corresponding to (or belong to) eigenvaluesThe characteristic polynomial of a 3.5.2 determinant (or) called a matrix (Note: the sub polynomial of a characteristic polynomial is) is a characteristic equation of a matrix:Let the order matrix be the unit matrix of the order, the eigenvalues of the matrix, and the matrixWith the elementary transformation, the upper triangular matrix can be obtained, and the product of the elements on the principal diagonal of the matrix is 0The value is the eigenvalue of the matrix.Example 7 uses the elementary transformation method of matrix to find the eigenvalues and eigenvectors of the matrixSolution:The product of the principal diagonal elements of the order is zero, i.e.EigenvalueThenTherefore, the corresponding eigenvectors areAll the corresponding eigenvectors are.WhenTherefore, the corresponding eigenvectors areThe entire feature vector at this time is.3.6, use the elementary transformation, and the two form is the standard typeTwo order homogeneous polynomials with variablesReferred to as the "yuan two times", referred to as the "twotimes".Order, rememberThen the two type can be expressed asA matrix of symmetric matrices of two order.When a series of elementary column transformations are applied to a matrix, the same elementary row transformation is applied to the block,When the block diagonal matrixWhen the child blocks are reduced, the. At this point, if the order, then into a standard shapeExamples are 8 and two times as standard.Solution: the quadratic matrix is twoImplementing elementary transformationIn this way, by coordinate transformation, of whichThe two form is a standard shapeNote: two types can be standardized in a variety of ways, and their standard shapes are not unique.Sum upTo solve some problems in algebra when using the elementary matrix transform can simplify the problem, such as the two type as the standard type, in addition to using elementary transformation method, also can be calculated using the orthogonal transformation method and collocation method, comparison of elementary transformation is simple, easy to calculate, easy to understand. The elementary transformation matrix has many applications in solving computational problems of linear algebra, the calculation format has many similar places, once mastered the operation of the matrix, we analyze and solve the equations of the ability will be greatly enhanced.In a word, the elementary transformation of matrix is an important method of calculation in linear algebra. We can use matrix elementary transformation to compute the rank of matrix, inverse matrix and matrix equation. With the development of science and technology, matrix has been applied to the natural, social, engineering, economic and other fields, and artificial intelligence, mobile phone communication and algorithm design and general analysis, the matrix in its application is communication optimization. We can not confine ourselves to the study of books. We should integrate theory with practice and make better use of theoretical knowledge to solve practical problems.Reference[1] 、 pre algebra group, Department of geometry and algebra, Department of mathematics, Peking University. Advanced Algebra (Third Edition), higher education press,.2003[2] Ma Juxia, Wu Yuntian. Linear Algebra (Second Edition). National Defense Industry Press.2009.8[3] Chen Zhizhong. Refined refining of linear algebra. Beijing Normal University press,.2006[4], Li Zhihui, Li Yongming. Typical problems and methods in advanced algebra. Science Press,.2008[5], Kang Yonghai, Zhu Baoyan. Application of elementary transformation of matrix in solving problems [J]. Journal of Songliao University (NATURAL SCIENCE EDITION),.1998 (3)[6], Yao Gang. Advanced Algebra (Second Edition) [M]., Fudan University press,.2008[7], Wang Junqing. On the application of elementary transformation in Higher Algebra [J]. Journal of Cangzhou Teachers College,.2002.18 (3)[8], Li Haiyan, Wang Yanfang. The whole course learning guide of Linear Algebra (three edition of the National People's Congress). Dalian University of Technology press,.2008.8Application of [9] Guangyan. Linear algebra lecture. Dalian University of Technology press.2008.7[10] Northwestern Polytechnical University advanced algebra compilation group. Advanced algebra. Science Press,.2008[11] Zhao Lixin, once Wencai. Characteristics of square matrix with elementary transformation of matrix valued.2004 mathematics [J]. University[12] Zhan Hua Lu, Lu established. Some applications of elementary transformation of higher mathematics of.2006.11 [J].。
数学专业英语词汇（M）
数学专业英语词汇(M)mach angle 马赫角mach cone 马赫锥mach number 马赫数machine computation 机破算machine computing 机破算machine equation 机平程machine language 机骑言machine word 计算机语mackey topology 麦基拓扑maclaurin expansion 马克劳林展开maclaurin formula 马克劳林公式macro instruction 广义指令macrooperation 大运算macroparameter 宏观参数macrostatistics 宏观统计学magic circle 幻圆magic cube 幻立方magic figure 幻图magic square 纵横图magnetic head 磁头magnetic store 磁存储器magnetic tape 磁带magnetohydrodynamics 磁铃力学magnitude 量main diagonal 衷角线main program 痔序major axis 长轴major cycle 大循环major premise 大前提major term 大词majorant 强级数majorant criterion 比较检验majorant series 强级数majority 多数majority decision function 多数判定函数majority function 强函数majority game 强对策majorized sequence 优化序列majorized series 优化级数mal posed problem 不适定问题malfunction 错误动作maltiple classification 廖分类manifold 廖manifold classification 廖分类manifold of flags 旗廖manifold without boundary 无边廖manipulation 操作mannheim curve 曼海姆曲线mantissa 尾数many body problem 多体问题many dimentional sepce 多维空间many valued composition law 多值合成律many valued function 多值函数many valued logic 多值逻辑many valued mapping 多值映射map 映射map coloring problem 地图着色问题map projection 地图投影mapping 映射mapping cone 映射锥mapping cylinder 映射柱mapping function 映射函数mapping norm 映射范数mapping of sets 集映射mapping of the boundary 边缘映射mapping space 映射空间mapping theorem 映射定理mapping transformation 映射变换marginal density 边缘密度marginal distribution 边缘分布marginal distribution density function 边缘分布密度函数marginal distribution function 边缘分布函数mark 记号market model 市场模型marking function 标记函数markoff chain 马尔可夫链markov chain 马尔可夫链markov decision process 马尔可夫决策过程markov matrix 马尔可夫矩阵markov process 马尔可夫过程markov transform 马尔可夫变换marriage problem 配对问题mass 质量master program 痔序master sample 标准样本matching 匹配matching theorems 匹配定理material implication 实质蕴涵mathematical 数学的mathematical analysis 数学分析mathematical approximation 数学近似法mathematical constant 数学常数mathematical expectation 期望值mathematical formula 数学公式mathematical induction 数学归纳法mathematical logic 数理逻辑mathematical model 数学模型mathematical pendulum 数学摆mathematical physics 数学物理mathematical programming 数学规划mathematical random sample 数学随机样本mathematical statistics 数理统计mathematics 数学mathieu equation 马提厄方程mathieu function 马提厄函数mathieu group 马提厄群matricial rank 矩阵的秩matrix 矩阵matrix algebra 矩阵代数matrix analysis 矩阵分析matrix calculation 矩阵计算matrix element 矩阵元matrix equation 矩阵方程matrix factorization method 矩阵因子分解方法matrix form 矩阵形式matrix function 矩阵函数matrix game 矩阵对策matrix group 矩阵群matrix inversion 矩阵求逆matrix norm 矩阵范数matrix of coefficients 系数矩阵matrix of the transformation 变换矩阵matrix operator 矩阵算子matrix power series 矩阵幂级数matrix product 矩阵积matrix representation 阵表示matrix ring 矩阵环matrix semigroup 矩阵半群matrix series 矩阵级数matrix solution 矩阵解matrix transformation 矩阵变换matrix tree theorem 矩阵狮理matrix unit 矩阵单位matroid 矩阵胚maximal abelian extension 最大阿贝耳扩张maximal chain 连通链maximal element 极大元maximal equivalent orber 极大整环maximal hermitian operator 最大埃尔米特算子maximal ideal 极大理想maximal ideal space 极大理想空间maximal operator 最大算子maximal order 极大整环maximal principle 最大值原理maximal separable extension 极大可分扩张maximal strip 极大带maximal tree 最大树生成树maximality 极大性maximin 极大极小maximization 极大化maximizing sequence 极大化序列maximum 最大maximum condition 极大条件maximum deviation 最大偏差maximum ergodic theorem 极大遍历定理maximum likelihood equations 极大似然方程maximum likelihood estimating function 极大似然估计量maximum likelihood estimator 极大似然估计量maximum likelihood method 极大似然法maximum likelihood principle 极大似然法maximum matching 极大匹配maximum modulus principle 最大模原理maximum number 最大数maximum of a function 函数最大maximum or minimum condition 极大或极小条件maximum point 最大点maximum principle 最大值原理maximum problem 极大值问题maximum solution 最大解maximum term 极大项maximum value 绝对极大值maxwell boltzmann distribution law 麦克斯韦玻耳兹曼分布律maxwell's distributlon law 麦克斯事分布律maxwell's equations 麦克斯事方程meager set 贫集mean 平均mean continuity 中数连续性mean convergence 平均收敛mean convergence of p th order p阶平均收敛mean curvature 平均曲率mean curvature of surface 曲面的平均曲率mean density 平均密度mean derivative 平均微商mean deviation 平均偏差mean difference 平均差mean error 平均误差mean life 平均寿命mean number 平均数mean ordinate 平均纵坐标mean pay off 平均支付mean proportional 比例中项mean square 均方mean square contingency 均方列联mean square deviation 方差mean square of error 误差的均方mean square value 均方值mean term 内项mean type 平均型mean value 平均值mean value method 平均值法mean value theorem 平均值定理mean vector 均值向量measurability 可测性measurable 可测的measurable function 可测函数measurable mapping 可测映射measurable set 可测集measure 测度measure of dispersion 离差的度量measure of skewness 偏度measure preserving transformation 保测变换measure space 测度空间measure theory 测度论measure zero 零测度measurement 测量measuring error 测量误差measuring rule 量尺mechanics 力学mechanism 机构median 中位数median line 中线median point 中点mediant 中间数medium 媒体meet 交meet homomorphic image 保交同态像meet irreducible element 交不可约元素mega 兆member 项member of an equation 方程的端边memory 存储器memory capacity 存储容量memory cell 存储单元memory register 存储寄存器mental arithmetic 心算meridian 子午线meromorphic differential 亚纯微分meromorphic function 亚纯函数meromorphic function element 亚纯函数元素meromorphic mapping 亚纯映射meromorphism 亚纯映射meromorphy 亚纯mesh point 网格点mesh size 网格大小mesokurtic distribution 常峰态分布meta axiom of choice 亚选择公理metabelian group 亚交换群metacompact space 亚紧空间metaharmonic function 亚低函数metalanguage 元语言metalogic of predicates 谓词元逻辑metatheorem 元定理meter 米method of approximation 近似法method of artificial variables 人工变量法method of balayage 扫除法method of characteristic curves 特者法method of comparison 比较法method of conjugate gradients 共轭梯度法method of difference 差分法method of elimination 消元法method of estimation 估计法method of exhaustion 穷竭法method of false position 试位法method of finite elements 有限元法method of fractional steps 分步法method of integration of partial differential equations 偏微分方程的积分法method of iteration 迭代法method of partial fractions 部分分数法method of perturbation 扰动法method of potentials 起脚石法method of power series 幂级数法method of principal axes 轴法method of principal components 种量法method of regularization 正则化法method of residues 剩余法method of runge kutta type 朗格库塔型的方法method of steepest ascent 最速上升法method of steepest descent 最速下降法method of successive approximation 逐次近似法method of undetermined coefficients 比较法metre 米metric 度量metric coefficient 度量系数metric connection 度量联络metric form 度量形式metric normal form of quadratic form 二次形式的度量标准形式metric space 度量空间metric subspace 度量子空间metric tensor 基本张量metric topology 度量拓扑metrically convex subset 度量凸子集metrically dense 度量的稠密metrizability 可度量性metrizable 可度量化的metrizable group 可度量化群metrizable uniform space 可度量化一致空间metrization 度量化metrization theorem of urysohn 乌里申度量化定理microlocal analysis 微局部分析mid square method 平方取中法middle term 中项midperpendicular 中垂线midpoint 中点midrange 中列数millimeter 毫米million 百万minimal automaton 极小自动机minimal basis 极小基minimal disjunctive normal form 极小析取范式minimal element 极小元素minimal generating set 不可约生成集minimal graph 极小图形minimal manifold 极小簇minimal model 极小模型minimal polynomial 极小多项式minimal propositional calculus 极小命题演算minimal solution 极小解minimal submanifold 极小子廖minimal sufficient estimator 最小充分估计量minimal sufficient statistic 最小充分统计量minimal surface 极小曲面minimal type 极小类型minimal variety 极小簇minimality 极小性minimax 极小极大minimax decision function 极小极大判决函数minimax inequality 极小极大不等式minimax principle 极小极大原理minimax solution 极小极大解minimax strategy 极小极大策略minimax theorem 极小极大定理minimization 极小化minimizing method 极小化法minimizing sequence 极小化序列minimum 最小minimum condition 极小条件minimum covering 极小覆盖minimum density 极小密度minimum integral 极小解minimum modulus 最小模minimum modulus principle 最小模原理minimum number 最小数minimum of a function 函数的最小minimum point 极小点minimum principle 极小原理minimum problem 极小问题minimum solution 极小解minimum value 最小值minimum with a condition 条件极小minimum with a constraint 条件极小minkowski approximation theorem 闵可夫斯基逼近定理minkowski inequalities 闵可夫斯基不等式minkowskian addition 闵可夫斯基加法minkowskian linear combination 闵可夫斯基线性组合minkowskian space 闵可夫斯基空间minor 子式minor arc 劣弧minor axis 短轴minor cycle 小循环minor determinant 子行列式minor premise 小前提minor term 小词minorant 弱函数minorant function 弱函数minuend 被减数minus infinity 负无穷大minus mark 负号minute 分miscalculation 计算误差;计算误差missing plot technique 缺区补救技术missing value 缺少值mistake 错误mix 混合mixed area 混合面积mixed concomitant 混合相伴式mixed differential parameter 混合微分参数mixed distribution 混合分布mixed graph 混合图形mixed group 混合群mixed ideal 混合理想mixed number 带分数mixed partial derivative 混合偏导数mixed problem 混合问题mixed side condition 混合边条件mixed strategy 混合策略mixed tensor 混合张量mixed type 混合型mixed vertex 混合顶点mixing problem 混合问题mixing ratio 混合比mixture 混合mnemonic 助记的mnemonic device 助记装置mnemonics 助记mobility 可动性modal class 众数组modal proposition 模态命题modal system 模态系统modal value 最常见的值modality logics 模态逻辑mode 众数mode of vibration 振动模式model 模型model of moving means 移动平均模型model test 模型试验model theory 模型理论modern geometry 近世几何学modification 变形modified bessel function 修正贝塞耳函数modified newton method 修正牛顿法modular 模的modular category 模范畴modular character 模特征modular equation 模方程modular figure 模图modular form 模形式modular function 模函数modular group 模群modular lattice 模格modular matrix 模矩阵modular substitution 模置换modular variety 模簇module 模module of boundaries 边界模module of homomorphisms 同态模module of program 程序的模module of quotients 商模moduli space 参模空间modulo 模modulus 绝对值;模modulus of a congruence 同余模modulus of elasticity 弹性模数modulus of periodicity 周期的模modulus of rigidity 刚性模量modus tollens 否定式moment 矩moment generating function 矩量母函数moment matrix 矩量矩阵moment of distribution 分布矩moment of force 力矩moment of inertia 惯性矩moment of momentum 动量矩moments method 矩量法momentum 动量monad 单子monge cone 蒙日锥monic 首一的monic polynomial 首一多项式monitor 监督程序monocyclic system 单循环系monodromy 单值monodromy group 单值群monodromy theorem 单值定理monogenic 单演的monogenic function 单演函数monogenic module 循环模monogyre 一次对称轴monoid 单式半群monoidal representation 单项表示monoidal transformation 单项变换monomial 单项式monomial equation 单项方程monomial factor 单项因子monomial form 单项形式monomial group 单项群monomial representation 单项表示monomorphism 单一同态monotone approximation 单灯近monotone class 单掂monotone decreasing 单递减的monotone decreasing function 单递减函数monotone function 单弹数monotone increasing function 单递增函数monotone numbering 单掂号monotone sequence 单凋列monotonic function 单弹数monotonic system of sets 单掂monotonic transformation 单典换monotonically decreasing sequence 单递减序列monotonically increasing sequence 单递增序列monotonicity 单翟monotony interval 单跌间monte carlo method 蒙特卡罗法monte carlo simulation 蒙特卡罗模拟montel space 空间moore smith convergence 穆尔史密斯收敛morphism 射morse inequalities 莫尔斯不等式morse theory 莫尔斯理论mortality rate 死亡率mortality table 死亡率表most powerful test 最大功效检定most probable duration 最可能持续时间most probable value 最常见的值most stringent test 最紧迫检验motion 运动motion equation 运动方程movable singularity 可移奇点move 步着movement 运动moving arm 移动臂moving average 移动平均moving average method 移动平均法moving frame 活动标架moving trihedral 怜三面形moving trihedron 怜三面形mrkoff process 马尔可夫过程mu continuity 连续性mu continuous function 连续函数mu integrable 可积分mu measurable 可测的mu singular 奇异multi dimensional integral 多维积分multi modal distribution 多重模态分布multi person game 多人对策multi phase sampling 多相抽样法multi purpose computer 万能计算机multi stage game 多阶段对策multi stage sampling 多级抽样法multi valued mapping 多值映射multi valuedness 多值性multiaddress 多地址multicollinearity 多重共线性multidimensional space 多维空间multigraph 多重图multigroup 超群multilinear form 多重线性形式multilinear function 多重线性函数multinomial 多项式multinomial coefficient 多项式系数multinomial distribution 多项分布multinomial expansion 多项展开式multinomial series 多项级数multinomial theorem 多项式定理multiple 倍数multiple arc 多重弧multiple correlation coefficient 多重相关系数multiple decision problem 多重判定问题multiple edge 多重棱multiple fourier series 多重傅里叶级数multiple hypergraph 多重超图multiple markov process 多重马尔可夫过程multiple point 多重点multiple regression 多重回归multiple root 多重根multiple sequence 多重序列multiple series 多重级数multiple stratification 多层化multiple tangent 多重切线multiple test 多重检验multiple valued 多值的multiple valued function 多值函数multiplicand 被乘数multiplicand register 被乘数寄存器multiplication 乘multiplication operator 乘法算子multiplication ring 乘环multiplication sign 乘号multiplication table 九九表multiplication theorem 乘法定理multiplicative 乘法的multiplicative axiom 乘法公理multiplicative character 乘法特贞multiplicative group 乘法群multiplicative lattice 乘格multiplicative process 繁殖过程multiplicatively closed set 积闭集multiplicator 乘数multiplicity 重数multiplicity of a root 根的重数multiplier 乘数;乘群multiplier register 乘数寄存器multiply 乘multiply connected domain 多连通区域multiply connected region 多连通区域multiply connected sequence 多连通序列multiply connected space 多连通空间multiply monotone sequence 多重单凋列multiply periodic function 多重周期函数multiply transitive group 多重可迁群multipolar 多极的multiprogram processing 多级程序处理multistage programming 多阶段规划multistep method 多步方法multitude 多数multivalent 多叶的multivalent function 多叶函数multivalued decision 多值判断multivariate analysis 多元分析multivariate analysis of variance 多元方差分析multivariate distribution 多元分布multivariate distribution function 多元分布函数multivariate statistics 多元统计multivector 多重矢量mutual information 交互信息mutually disjoint 互不相交的mutually disjoint events 互不相交事件mutually disjoint subsets 互不相交子集mutually independent events 互相独立事件myria 万myriad 无数的myriads 无数。
矩阵和的Schur补的性质
矩阵和的Schur补的性质狄勇婧;段复建【摘要】In order to reduce the large-scale matrix calculation and simplify the matrix equation numerical calculation, the properties of Schur complement of the sum of block matrix was studied. The two properties of Schur complement of the sum of block matrix were obtained by the effect of Schur complement with the replacement of block. They were proved in the theory, the theoretical support was provided to deal with the large-scale matrix calculation.%为减少大规模的矩阵计算,简化矩阵方程的数值计算,研究了分块矩阵和的Schur补的性质.通过研究矩阵的分块置换对Schur补的影响,获得分块矩阵和的Schur补的2个性质,并在理论上予以证明,为处理大规模的矩阵计算提供了理论支撑.【期刊名称】《桂林电子科技大学学报》【年(卷),期】2012(032)006【总页数】3页(P490-492)【关键词】分块矩阵;分块置换;Schur补【作者】狄勇婧;段复建【作者单位】桂林电子科技大学数学与计算科学学院,广西桂林541004【正文语种】中文【中图分类】O151.21矩阵Schur补的概念由Issai Schur［1］提出，随着对矩阵理论的研究与发展，Schur补公式的提出大大地推动了数学领域的发展，并且这个公式一直延用至今。
许多人利用Schur补的相关概念进行科学研究，并获得了丰富的科研成果。
传递矩阵法在等截面梁计算中的应用
在 １９２０ 年 Ｌｒｉｅ Ｔ 等最早提出传递矩阵的概念ꎬ最初该 方法是用于弹性构件所产生的一维线形震动问题上ꎬ它最主 要的思想是将一个结构的力学行为的分析分解为很多个微 小结构的力学行为过程ꎬ然后再将这些微小结构的力学行为 过程“拼接”起来ꎬ这就达到了传递的过程ꎮ 对这些结构的微 分方程进行求解转化成各种矩阵ꎬ然后通过计算机把这些矩 阵相乘ꎬ从而达到分析结构内力和力学性能的目的ꎮ
收稿日期:２０１８ － ０４ － ０８ 作者简介:沈宏任(１９９２ － ) ꎬ男ꎬ四川资中人ꎬ硕士研究生ꎬ主要研究 方向:桥梁工程ꎮ 基金项目:贵州省交通运输厅资助项目(２０１５ － １２３ － ０４１) ꎻ贵州省科 技厅项目( 黔科合支撑[２０１６]２０４７)
不同的问题和要求ꎬ写出计算截面的矩阵向量ꎬ然后根据各 个单元之间的传递关系ꎬ得出与之相对应的传递矩阵ꎬ再结 合结构的边界条件进行求解ꎬ最后得出该结构的内力和位移 等我们所需要的结果ꎮ 它的求解过程如图 １ 所示ꎮ
Ｔｈｅ Ｔｒａｎｓｆｅｒ Ｍａｔｒｉｘ Ｍｅｔｈｏｄ ｉｓ Ａｐｐｌｉｅｄ ｉｎ ｔｈｅ Ｃａｌｃｕｌａｔｉｏｎ ｏｆ Ｅｑｕａｌ Ｓｅｃｔｉｏｎ Ｂｅａｍ
ＳＨＥＮ ＨｏｎｇｒｅｎꎬＤＡＩ Ｘｉｊｕｎ ( Ｓｃｈｏｏｌ ｏｆ Ｃｉｖｉｌ Ｂｕｉｌｄｉｎｇ ａｎｄ ｔｈｅ Ｅｎｖｉｒｏｎｍｅｎｔꎬ ＸｉＨｕａ ＵｎｉｖｅｒｓｉｔｙꎬＣｈｅｎｇｄｕ ６１００３９ꎬＣｈｉｎａ) Ａｂｓｔｒａｃｔ: Ｔｈｅ ｔｒａｎｓｆｅｒ ｍａｔｒｉｘ ｉｓ ａ ｋｉｎｄ ｏｆ ｓｔｒｕｃｔｕｒａｌ ａｎａｌｙｓｉｓ ｍｅｔｈｏｄ ｔｈａｔ ｈａｓ ａｒｉｓｅｎ ｉｎ Ｃｈｉｎａ ｗｉｔｈ ｔｈｅ ｄｅｖｅｌｏｐｍｅｎｔ ｏｆ ｃｏｍｐｕｔｅｒｓ. Ｂａｓｅｄ ｏｎ ｔｈｅ ｐｒｅｃｉｓｅ ｓｏｌｕｔｉｏｎ ｏｆ ｔｈｅ ｓｔｒｕｃｔｕｒｅ ｃｏｎｔｒｏｌ ｄｉｆｆｅｒｅｎｔｉａｌ ｅｑｕａｔｉｏｎ ａｎｄ ｔｈｅ ｕｓｅ ｏｆ ｔｈｅ ｃｏｍｐｕｔｅｒꎬａｎｄ ｔｈｅ ｓｉｍｐｌｅ ｍｕｌｔｉｐｌｉｃａｔｉｏｎ ｏｆ ｔｈｅ ｍａｔｒｉｘ ｉｓ ｕｓｅｄ ｔｏ ａｎａｌｙｚｅ ｔｈｅ ｓｔａｔｉｃꎬｄｙｎａｍｉｃ ａｎｄ ｓｔａｂｉｌｉｔｙ ｏｆ ｔｈｅ ｓｔｒｕｃｔｕｒｅ. Ｍａｔｌａｂ ｐｒｏｇｒａｍ ｈａｓ ｇｒｅａｔ ａｄｖａｎｔａｇｅｓ ｉｎ ｎｕｍｅｒｉｃａｌ ｃａｌｃｕｌａｔｉｏｎꎬｅｓｐｅｃｉａｌｌｙ ｍａｔｒｉｘ ｃａｌｃｕｌａｔｉｏｎ ｉｓ ｓｉｍｐｌｅｒ ｔｈａｎ ｏｔｈｅｒ ｐｒｏｇｒａｍｍｉｎｇ ｓｏｆｔｗａｒｅ. Ｉｎ ｔｈｉｓ ｐａｐｅｒꎬ ｗｅ ｂｒｉｅｆｌｙ ｉｎｔｒｏｄｕｃｅ ｔｈｅ ｃａｌｃｕｌａｔｉｏｎ ｏｆ ｔｈｅ ｔｒａｎｓｆｅｒ ｍａｔｒｉｘ ｏｆ ｃｒｏｓｓ － ｓｅｃｔｉｏｎ ｓｔｒａｉｇｈｔ ｂｅａｍ ｗｉｔｈ Ｍａｔｌａｂ. Ｋｅｙ ｗｏｒｄｓ:ｍａｔｌａｂꎻｔｈｅ ｔｒａｎｓｆｅｒ ｍａｔｒｉｘ ꎻｃｏｍｐｕｔａｔｉｏｎｓ
riscv 矩阵乘法 汇编代码
riscv 矩阵乘法汇编代码RISC-V 是一个基于精简指令集（RISC）的开源指令集架构（ISA）。
矩阵乘法是一个常见的计算密集型任务，通常用于线性代数、图形学、机器学习等领域。
然而，直接写出用于矩阵乘法的RISC-V 汇编代码是相当复杂且冗长的，因为它涉及到循环、内存访问和算术运算。
此外，RISC-V 指令集本身并不直接支持矩阵乘法这样的高级操作，因此需要通过一系列的基础指令来实现。
下面是一个非常简化的示例，展示了如何使用RISC-V 汇编语言实现两个2x2 矩阵的乘法。
请注意，这只是一个教学示例，实际的矩阵乘法实现可能会更加复杂，并且会考虑性能优化。
assembly.section .datamatrixA: .word 1, 2, 3, 4 # 2x2 matrix: [[1, 2], [3, 4]]matrixB: .word 5, 6, 7, 8 # 2x2 matrix: [[5, 6], [7, 8]]result: .word 0, 0, 0, 0 # Space for the result matrix.section .text.globl _start_start:# Load matrix A and B into registersla a0, matrixA # Load address of matrix A into a0la a1, matrixB # Load address of matrix B into a1la a2, result # Load address of result matrix into a2# Initialize loop countersli t0, 0 # i = 0li t1, 0 # j = 0li t2, 0 # k = 0matrix_multiply:# Check if we've finished all rows of Abge t0, 2, end_multiply# Check if we've finished all columns of Bbge t1, 2, next_row_a# Inner loop: calculate a single element of the result matrixli t3, 0 # Accumulator for the element calculationli t4, 0 # l = 0 (for the inner loop)inner_loop:# Check if we've finished the inner loopbge t4, 2, next_column_b# Load elements of A and Blw t5, 0(a0) # Load element A[i][l] into t5lw t6, 0(a1) # Load element B[l][j] into t6# Calculate A[i][l] * B[l][j]mul t7, t5, t6 # t7 = A[i][l] * B[l][j]# Accumulate the resultadd t3, t3, t7 # Accumulator += A[i][l] * B[l][j]# Increment the inner loop counter and pointersaddi t4, t4, 1 # l++addi a0, a0, 4 # Move to the next element in the same row of Aaddi a1, a1, 4 # Move to the next element in the same column of B# Jump back to the beginning of the inner loopj inner_loopnext_column_b:# Reset the inner loop counter and pointersli t4, 0 # l = 0addi a0, a0, -8 # Move back to the beginning of the current row of Aaddi a1, a1, 8 # Move to the next column of B# Increment the column counter of Baddi t1, t1, 1 # j++# Jump back to the beginning of the row loopj matrix_multiplynext_row_a:# Reset the column counter of B and pointersli t1, 0 # j = 0addi a0, a0, 8 # Move to the next row of Ala a1, matrixB # Reset the pointer to the beginning of B# Increment the row counter of Aaddi t0, t0, 1 # i++# Jump back to the beginning of the matrix multiplicationj matrix_multiplyend_multiply:# Store the result back to memoryla a0, resultsw t3, 0(a0) # This is just a placeholder. In reality, you would need to store each element of the result matrix.# Exit the programli a7, 10 # System call code for exitli a0, 0 # Exit statusecall # Perform the system call注意：这个代码只是一个非常简化的示例，它并没有正确地计算和存储整个结果矩阵。
任意椭球内的均匀分布随机变量的协方差
任意椭球内的均匀分布随机变量的协方差1.协方差可以衡量两个随机变量之间的线性关系强度。
The covariance can measure the strength of the linear relationship between two random variables.2.当协方差为正时，表示两个变量同向变化。
When the covariance is positive, it indicates that the two variables change in the same direction.3.负协方差意味着两个变量呈现反向变化。
Negative covariance means that the two variables change in opposite directions.4.协方差为零说明两个变量之间没有线性关系。
A covariance of zero indicates that there is no linear relationship between the two variables.5.协方差的绝对值越大，代表两个变量之间的关联越强。
The larger the absolute value of the covariance, the stronger the relationship between the two variables.6.通过计算协方差，可以了解随机变量之间的关系及变化趋势。
By calculating the covariance, we can understand the relationship and trend of change between random variables.7.协方差的单位是两个变量的乘积的单位。
The unit of covariance is the product of the units of the two variables.8.在椭球内的均匀分布随机变量的协方差可以通过数学公式来计算。
c++矩阵运算库函数
c++矩阵运算库函数C++是一种功能强大的编程语言，可以用于开发各种应用程序和计算机系统。
在科学计算和数据处理领域，矩阵运算是非常常见的操作之一。
为了简化矩阵运算的代码编写过程，提高开发效率，许多矩阵运算库函数都被开发出来。
以下是一些常见的C++矩阵运算库函数及其使用方法的参考内容。
1. Eigen库：Eigen是一个开源的C++模板库，提供了许多用于线性代数和矩阵运算的函数和类。
它使用模板元编程技术实现了高性能的矩阵运算，支持动态大小的矩阵和固定大小的矩阵。
以下是Eigen库的一个简单示例：```#include <iostream>#include <Eigen/Dense>int main() {Eigen::MatrixXd m(2,2);m(0,0) = 1.0;m(0,1) = 2.0;m(1,0) = 3.0;m(1,1) = 4.0;Eigen::VectorXd v(2);v(0) = 1.0;v(1) = 2.0;Eigen::VectorXd result = m * v;std::cout << result << std::endl;return 0;}```在上面的示例中，我们首先包含了Eigen的头文件，并定义了一个2x2的矩阵m和一个大小为2的向量v。
然后，我们使用矩阵乘向量的方式计算了结果，并输出了结果。
2. Armadillo库：Armadillo是一个用于线性代数和科学计算的C++库，提供了高级的矩阵和向量操作。
它具有简单易用的接口和高性能的计算能力。
以下是Armadillo库的一个简单示例：```#include <iostream>#include <armadillo>int main() {arma::mat A = {{1.0, 2.0}, {3.0, 4.0}};arma::vec b = {1.0, 2.0};arma::vec result = A * b;std::cout << result << std::endl;return 0;}```在上面的示例中，我们首先包含了Armadillo的头文件，并定义了一个2x2的矩阵A和一个大小为2的向量b。
矩阵合同变换的应用
矩阵合同变换的应用Matrix congruence transformation is an important concept in mathematics with various applications in different fields. It involves the transformation of a matrix through multiplication by an invertible matrix, which results in a new matrix with similar properties. This concept is widely used in linear algebra, computer graphics, and physics, among other disciplines. The application of matrix congruence transformation can lead to simplified calculations, improved visualization, and better understanding of complex systems and structures.矩阵合同变换是数学中的一个重要概念，在不同领域具有各种应用。
它涉及通过乘以可逆矩阵对一个矩阵进行变换，从而得到一个具有相似性质的新矩阵。
这个概念在线性代数、计算机图形学和物理等领域被广泛应用。
矩阵合同变换的应用可以简化计算、改善可视化效果，并更好地理解复杂系统和结构。
In linear algebra, matrix congruence transformation is used to simplify calculations involving large matrices. By transforming a matrix into a congruent form, it becomes easier to performoperations such as matrix multiplication, inversion, and determinant calculation. This simplification can be particularly helpful in solving systems of linear equations, finding eigenvalues and eigenvectors, and studying transformations in vector spaces. The ability to transform matrices through congruence allows for more efficient and accurate computations in various mathematical applications.在线性代数中，矩阵合同变换被用来简化涉及大矩阵的计算。
abaqus输出单元刚度矩阵
abaqus输出单元刚度矩阵Abaqus is a powerful finite element analysis software widely used in the field of structural mechanics. In the process of analyzing structural behavior, one of the crucial aspects is the calculation of the stiffness matrix of each finite element. The stiffness matrix represents the relationship between the forces and displacements in a linear elastic system, and its calculation is a fundamental step in solving the equations of motion for a structure.In Abaqus, the stiffness matrix is automatically calculated during the analysis process. It is essential to obtain the stiffness matrix accurately as it informs us about the response of the structure under various loading conditions. In this article, I will explain the process of obtaining the stiffness matrix in Abaqus, step by step.Step 1: Creating the GeometryBefore proceeding with the calculation of the stiffness matrix, it is necessary to create the geometry of the structure in Abaqus. This involves defining the dimensions, shape, and boundary conditions of the model. Abaqus provides various tools and techniques for creating the geometry, including meshing, surfaces, profiles, andsections.Step 2: Assigning Material PropertiesAfter creating the geometry, the next step is to assign material properties to the model. Material properties include parameters such as Young's modulus, Poisson's ratio, and density, which define the behavior of the material under loading. Abaqus supports a range of material models, including linear elastic, plastic, and viscoelastic materials. The accurate assignment of material properties is crucial for obtaining an accurate stiffness matrix.Step 3: Defining Boundary ConditionsBoundary conditions play a vital role in determining the behavior of a structure. In Abaqus, boundary conditions are defined by specifying constraints on the model. These constraints can be in the form of fixed displacements, applied forces, or thermal effects. It is important to define the boundary conditions correctly to ensure that the stiffness matrix accurately reflects the behavior of the structure under different loading conditions.Step 4: Selecting Element TypesAbaqus provides a wide range of element types to simulate different types of structures and loading conditions. The selection of element types depends on factors such as the geometry of the structure, material properties, and the desired level of accuracy. Each element type has its own stiffness matrix formulation, and it is essential to choose the appropriate element type for accurate results.Step 5: Meshing and Element ConnectivityMeshing is the process of dividing the geometry into a finite number of elements. Abaqus provides several meshing techniques, including structured and unstructured meshing. Meshing the structure correctly is crucial for obtaining an accurate stiffness matrix. Once the mesh is created, the elements are connected to each other based on their connectivity. The element connectivity information is then used to derive the global stiffness matrix for the entire structure.Step 6: Applying Loads and Solving the System of EquationsAfter the geometry, material properties, boundary conditions, element types, and meshing are defined, the next step is to apply the loads to the structure. These loads can be in the form of applied forces, pressure, temperature, or any other type of loading. Once the loads are applied, Abaqus solves the system of equations to obtain the displacements of the nodes and the corresponding reaction forces. The stiffness matrix is calculated during the solution of the system of equations.Step 7: Extracting the Stiffness MatrixOnce the analysis is complete, Abaqus provides various methods to extract the stiffness matrix from the results. The stiffness matrix is typically represented in a matrix form, where each element corresponds to the stiffness between two degrees of freedom. Extracting the stiffness matrix allows us to understand the behavior of the structure more precisely and use it for various purposes, such as modal analysis, structural optimization, and design verification.In conclusion, obtaining the stiffness matrix is an essential step in structural analysis using Abaqus. It is a complex process involvingmultiple steps, including geometry creation, material assignment, boundary condition definition, meshing, element type selection, load application, solution, and stiffness matrix extraction. By accurately following these steps, engineers can obtain the stiffness matrix of a structure in Abaqus and use it for various applications to ensure the structural integrity and safety.。
cuda 共享内存 矩阵乘法
cuda 共享内存矩阵乘法英文回答：CUDA shared memory is a valuable resource that can greatly improve the performance of matrix multiplication operations. In matrix multiplication, each element of the resulting matrix is calculated by taking the dot product of a row from the first matrix and a column from the second matrix. This operation involves a lot of memory accesses, which can be a bottleneck in terms of performance.Shared memory is a small but fast memory space that is shared among threads within a block. It is located on the chip and has much lower latency compared to global memory. By utilizing shared memory, we can reduce the number of global memory accesses and improve the overall performance of matrix multiplication.To perform matrix multiplication using shared memory, we can divide the matrices into smaller submatrices andload them into shared memory. Each thread in a block is responsible for calculating a single element of the resulting matrix. By loading the required submatrices into shared memory, we can minimize the number of global memory accesses and increase the data reuse within a block.Here's a step-by-step explanation of how shared memory can be used in matrix multiplication:1. Divide the matrices into smaller submatrices: Divide the input matrices into smaller submatrices that can fit into shared memory. For example, if the input matrices are of size N x N, we can divide them into submatrices of size K x K, where K is the size that can fit into shared memory.2. Load submatrices into shared memory: Each thread ina block loads a submatrix from the input matrices into shared memory. The submatrix can be loaded row by row or column by column, depending on the access pattern required for the calculation.3. Synchronize threads: After loading the submatricesinto shared memory, we need to synchronize the threads within a block to ensure that all the required data is available in shared memory before proceeding with the calculation. This can be done using the __syncthreads() function.4. Perform matrix multiplication: Each thread calculates a single element of the resulting matrix by taking the dot product of a row from the first submatrix and a column from the second submatrix, both stored in shared memory. The result is then accumulated in a local variable.5. Synchronize threads again: After the calculation is done, we need to synchronize the threads again to ensure that all the threads have finished their calculations before proceeding to the next step.6. Store the result: Each thread stores its calculated result into the output matrix, which is stored in global memory.7. Repeat the process: If the size of the input matrices is larger than the size that can fit into shared memory, we need to repeat the process for the remaining submatrices until all the elements of the resulting matrix are calculated.Using shared memory in matrix multiplication can significantly improve the performance compared to using only global memory. The reduction in global memory accesses and the increase in data reuse within a block can lead to a faster execution time.中文回答：CUDA共享内存是一个宝贵的资源，可以极大地提高矩阵乘法操作的性能。
MatCalc----矩阵计算
Notes on Matrix CalculusPaul L.Fackler∗North Carolina State UniversitySeptember27,2005Matrix calculus is concerned with rules for operating on functions of matrices.For example,suppose that an m×n matrix X is mapped into a p×q matrix Y.We are interested in obtaining expressions for derivatives such as∂Y ij∂X kl,for all i,j and k,l.The main difficulty here is keeping track of where things are put.There is no reason to use subscripts;it is far better instead to use a system for ordering the results using matrix operations.Matrix calculus makes heavy use of the vec operator and Kronecker products.The vec operator vectorizes a matrix by stacking its columns(it is convention that column rather than row stacking is used).For example, vectorizing the matrix12 34 56∗Paul L.Fackler is an Associate Professor in the Department of Agricultural and Re-source Economics at North Carolina State University.These notes are copyrighted mate-rial.They may be freely copied for individual use but should be appropriated referenced in published work.Mail:Department of Agricultural and Resource EconomicsNCSU,Box8109Raleigh NC,27695,USAe-mail:paul fackler@Web-site:/∼pfackler/c 2005,Paul L.Fackler1produces135246The Kronecker product of two matrices,A and B,where A is m×n and B is p×q,is defined asA⊗B=A11B A12B...A1n BA21B A22B...A2n B............A m1B A m2B...A mn B,which is an mp×nq matrix.There is an important relationship between the Kronecker product and the vec operator:vec(AXB)=(B ⊗A)vec(X).This relationship is extremely useful in deriving matrix calculus results.Another matrix operator that will prove useful is one related to the vec operator.Define the matrix T m,n as the matrix that transforms vec(A)into vec(A ):T m,n vec(A)=vec(A ).Note the size of this matrix is mn×mn.T m,n has a number of special properties.Thefirst is clear from its definition;if T m,n is applied to the vec of an m×n matrix and then T n,m applied to the result,the original vectorized matrix results:T n,m T m,n vec(A)=vec(A).ThusT n,m T m,n=I mn.The fact thatT n,m=T−1m,nfollows directly.Perhaps less obvious is thatT m,n=T n,m2(also combining these results means that T m,n is an orthogonal matrix).The matrix operator T m,n is a permutation matrix,i.e.,it is composed of0s and1s,with a single1on each row and column.When premultiplying another matrix,it simply rearranges the ordering of rows of that matrix (postmultiplying by T m,n rearranges columns).The transpose matrix is also related to the Kronecker product.With A and B defined as above,B⊗A=T p,m(A⊗B)T n,q.This can be shown by introducing an arbitrary n×q matrix C:T p,m(A⊗B)T n,q vec(C)=T p,m(A⊗B)vec(C )=T p,m vec(BC A )=vec(ACB )=(B⊗A)vec(C).This implies that((B⊗A)−T p,m(A⊗B)T n,q)vec(C)=0.Because C is arbitrary,the desired result must hold.An immediate corollary to the above result is that(A⊗B)T n,q=T m,p(B⊗A).It is also useful to note that T1,m=T m,1=I m.Thus,if A is1×n then (A⊗B)T n,q=(B⊗A).When working with derivatives of scalars this can result in considerable simplification.Turning now to calculus,define the derivative of a function mapping n→ m as the m×n matrix of partial derivatives:[Df]ij=∂f i(x)∂x j.For example,the simplest derivative isdAxdx=A.Using this definition,the usual rules for manipulating derivatives apply naturally if one respects the rules of matrix conformability.The summation rule is obvious:D[αf(x)+βg(x)]=αDf(x)+βDg(x),3whereαandβare scalars.The chain rule involves matrix multiplication, which requires conformability.Given two functions f: n→ m and g: p→ n,the derivative of the composite function isD[f(g(x))]=f (g(x))g (x).Notice that this satisfies matrix multiplication conformability,whereas the expression g (x)f (g(x))attempts to postmultipy an n×p matrix by an m×n matrix.To define a product rule,consider the expression f(x) g(x), where f,g: n→ m.The derivative is the1×n vector given byD[f(x) g(x)]=g(x) f (x)+f(x) g (x).Notice that no other way of multiplying g by f and f by g would ensure conformability.A more general version of the product rule is defined below.The product rule leads to a useful result about quadratic functions:dx Axdx=x A+x A =x (A+A ).When A is symmetric this has the very natural form dx Ax/dx=2x A.These rules define derivatives for vectors.Defining derivatives of matri-ces with respect to matrices is accomplished by vectorizing the matrices,so dA(X)/dX is the same thing as d vec(A(X))/d vec(X).This is where the the relationship between the vec operator and Kronecker products is useful. Consider differentiating dx Ax with respect to A(rather than with respect to x as above):d vec(x Ax) d vec(A)=d(x ⊗x )vec(A)d vec(A)=(x ⊗x )(the derivative of an m×n matrix A with respect to itself is I mn).A more general product rule can be defined.Suppose that f: n→m×p and g: n→ p×q,so f(x)g(x): n→ m×ing the relationship between the vec and Kronecker product operatorsvec(I m f(x)g(x)I q)=(g(x) ⊗I m)vec(f(x))=(I q⊗f(x))vec(g(x)).A natural product rule is thereforeDf(x)g(x)=(g(x) ⊗I m)f (x)+(I q⊗f(x))g (x).This can be used to determine the derivative of dA A/dA where A ism×n.vec(A A)=(I n⊗A )vec(A)=(A ⊗I n)vec(A )=(A ⊗I n)T m,n vec(A).4Thus(using the product rule)dA AdA=(I n⊗A )+(A ⊗I n)T m,n.This can be simplified somewhat by noting that(A ⊗I n)T m,n=T n,n(I n⊗A ).ThusdA AdA=(I n2+T n,n)(I n⊗A ).The product rule is also useful in determining the derivative of a matrix inverse:dA−1A dA =(A ⊗I n)dA−1dA+(I n⊗A−1).But A−1A is identically equal to I,so its derivative is identically0.ThusdA−1dA=−(A ⊗I n)−1(I n⊗A−1)=−(A− ⊗I n)(I n⊗A−1)=−(A− ⊗A−1).It is also useful to have an expression for the derivative of a determinant. Suppose A is n×n with|A|=0.The determinant can be written as theproduct of the i th row of the adjoint of A(A∗)with the i th column of A: |A|=A∗i·A·i.Recall that the elements of the i th row of A∗are not influenced by the elements in the i th column of A and hence∂|A|∂A·i=A∗i·.To obtain the derivative with respect to all of the elements of A,we can concatenate the partial derivatives with respect to each column of A:d|A| dA =[A∗1·A∗2·...A∗n·]=|A|[A−1]1·[A−1]2·...[A−1]n·=|A|vecA−.The following result is an immediate consequenced ln|A| dA =vecA−.5Matrix differentiation results allow us to compute the derivatives of the solutions to certain classes of equilibrium problems.Consider,for example, the solution,x,to a linear complementarity problem LCP(M,q)that solvesMx+q≥0,x≥0,x (Mx+q)=0The i th element of x is either exactly equal to0or is equal to the i th element of Mx+q.Define a diagonal matrix D such that D ii=1if x>0and equal 0otherwise.The solution can then be written as x=−ˆM−1Dq,where ˆM=DM+I−D.It follows that∂x∂q=−ˆM−1Dand that∂x ∂M =∂x∂ˆM−1∂ˆM−1∂ˆM∂ˆM∂M=(−q D⊗I)(−ˆM− ⊗ˆM−1)(I⊗D) =q DˆM− ⊗ˆM−1D=x ⊗∂x/∂qGiven the prevalence of Kronecker products in matrix derivatives,it would be useful to have rules for computing derivatives of Kronecker prod-ucts themselves,i.e.dA⊗B/dA and dA⊗B/dB.Because each element of a Kronecker product involves the product of one element from A multiplied by one element of B,the derivative dA⊗B/dA must be composed of zeros and the elements of B arranged in a certain fashion.Similarly,the derivative dA⊗B/dB is composed of zeros and the elements of A arranged in a certain fashion.6It can be verified that dA⊗B/dA can be written asdA⊗B dA =Ψ10 0Ψ20 0............Ψq0 00Ψ1 00Ψ2 0............0Ψq 0............00...Ψ100...Ψ2............00...Ψq=I n⊗Ψ1Ψ2...ΨqwhereΨi=B1i0 0B2i0 0............B pi0 00B1i 00B2i 0............0B pi 0............00...B1i00...B2i............00...B pi=I m⊗B·iThis can be written more compactly asdA⊗BdA=(I n⊗T qm⊗I p)(I mn⊗vec(B))=(I nq⊗T mp)(I n⊗vec(B)⊗I m).7Similarly dA⊗B/dB can be written asdA⊗B dB =Θ10 00Θ1 0............00...Θ1Θ20 00Θ2 0............00...Θ2............Θq0 00Θq 0............00...ΘqwhereΘi=A i10 00A i1 0............00...A i1A i20 00A i2 0............00...A i2............A im0 00A im 0............00...A im.This can be written more compactly asdA⊗BdB=(I n⊗T qm⊗I p)(vec(A)⊗I pq)=(T pq⊗I mn)(I q⊗vec(A)⊗I p). Notice that if either A is a row vector(m=1)or B is a column vector(q=1),the matrix(I n⊗T qm⊗I p)=Imnpq and hence can be ignored.To illustrate a use for these relationships,consider the second derivativeof xx with respect to x,an n-vector.dxxdx=x⊗I n+I n⊗x;8henced2xxdxdx=(T nn⊗I n)(I n⊗vec(I n)+vec(I n)⊗I n).Another example isd2A−1 dAdA =(I n⊗T nn⊗I n)I n2⊗vecA−1T nn+vecA−⊗I n2A− ⊗A−1)Often,especially in statistical applications,one encounters matrices that are symmetrical.It would not make sense to take a derivative with respect to the i,j th element of a symmetric matrix while holding the j,i th element constant.Generally it is preferable to work with a vectorized version of a symmetric matrix that excludes with the upper or lower portion of the matrix.The vech operator is typically taken to be the column-wise vector-ization with the upper portion excluded:vech(A)=A11···A n,1A22···A n2···A nn−1A nn.One obtains this by selecting elements of vec(A)and therefore we can write vech(A)=S n vec(A),where S n is an n(n+1)/2×n2matrix of0s and1s,with a single1in each row.The vech operator can be applied to lower triangular matrices as well;there is no reason to take derivatives with respect to the upper part of a lower triangular matrix(it can also be applied to the transpose of an upper triangular matrix).The use of the vech operator is also important in efficient computer storage of symmetric and triangular matrices.To illustrate the use of the vech operator in matrix calculus applications, consider an n×n symmetric matrix C defined in terms of a lower triangular matrix,L,C=LL .9Using the already familiar methods it is easy to see thatdCdL=(I n⊗L)T n,n+(L⊗I n).Using the chain ruled vech(C) d vech(L)=d vech(C)dCdCdLdLd vech(L)=S ndCdLS n.Inverting this expression provides an expression for d vech(L)/d vech(C).Related to matrix derivatives is the issue of Taylor expansions of matrix-to-matrix functions.One way to think of matrix derivatives is in terms of multidimensional arrays.An mn×pq matrix can also be thought of as an m×n×p×q4-dimensional array.The“reshape”function in MATLAB implements such transformations.The ordering of the individual elements has not change,only the way the elements are indexed.The d th order Taylor expansion of a function f(X):R m×n→R p×q at ˜X can be computed in the following wayf=vec(f(˜X))dX=vec(X−˜X)for i=1to d{f i=f(i)(˜X)dXfor j=2to if i=reshape(f i,mn(pq)j−2,pq)dX/jf=f+f i}f=reshape(f,m,n)The techniques described thus far can be applied to computation of derivatives of common“special”functions.First,consider the derivative of a nonnegative integer power A i of a square matrix A.Application of the chain rule leads to the recursive definitiondA i dA =dA i−1AdA=(A ⊗I)dA i−1dA+(I⊗A i−1)which can also be expressed as a sum of i termsdA i dA =ij=1(A )i−j⊗A j−1.10This result can be used to derive an expression for the derivative of the matrix exponential function,which is defined in the usual way in terms of a Taylor expansion:exp(A)=∞i=0A ii!.Thusd exp(A)dA =∞i=01i!ij=1(A )i−j⊗A j−1.The same approach can be applied to the matrix natural logarithm:ln(A)=−∞i=11i(I−A)i.11Summary of Operator ResultsA is m×n,B is p×q,X is defined comformably.A⊗B=A11B A12B...A1n BA21B A22B...A2n B............A m1B A m2B...A mn B(AC⊗BD)=(A⊗B)(C⊗D) (A⊗B)−1=A−1⊗B−1(A⊗B) =A ⊗Bvec(AXB)=(B ⊗A)vec(X) trace(AX)=vec(A ) vec(X) trace(AX)=trace(XA)T m,n vec(A)=vec(A )T n,m T m,n=I mnT n,m=T−1m,nT m,n=T n,mB⊗A=T p,m(A⊗B)T n,q12Summary of Differentiation ResultsA is m×n,B is p×q,x is n×1,X is defined comformably.[Df]ij=d f i(x) dx jdAxdx=AD[αf(x)+βg(x)]=αDf(x)+βDg(x)D[f(g(x))]=f (g(x))g (x)D[f(x)g(x)]=(g(x) ⊗I m)f (x)+(I p⊗f(x))g (x). dx Axdx=x (A+A )d vec(x Ax)d vec(A)=(x ⊗x )dA AdA=(I n2+T n,n)(I n⊗A )dAAdA=(I m2+T m,m)(A⊗I m)dx A AxdA=2x ⊗x A (when x is a vector) dx AA xdA=2x A⊗x (when x is a vector) dAXBdX=B ⊗AdA−1dA=−(A− ⊗A−1)d ln|A|dA=vec(A− )d trace(AX)dX=vec(A )13dA⊗BdA=(I n⊗T qm⊗I p)(I mn⊗vec(B))=(I nq⊗T mp)(I n⊗vec(B)⊗I m). dA⊗BdB=(I n⊗T qm⊗I p)(vec(A)⊗I pq)=(T pq⊗I mn)(I q⊗vec(A)⊗I p).dxxdx=x⊗I n+I n⊗xd2xxdxdx=(T nn⊗I n)(I n⊗vec(I n)+vec(I n)⊗I n)dA i dA =(A ⊗I)dA i−1dA+(I⊗A i−1)=ij=1(A )i−j⊗A j−1.d exp(A)dA =∞i=01i!ij=1(A )i−j⊗A j−1.14。
matrix计算
matrix计算Matrix是一种数学工具，它在计算机科学、物理学、统计学等领域都有广泛应用。
本文将介绍Matrix的基本概念、运算规则以及在不同领域中的应用。
一、Matrix的基本概念Matrix，中文翻译为“矩阵”，是一个由数个数按照矩形排列成的方阵。
矩阵的行数和列数分别决定了它的维度。
例如，一个2×3的矩阵有2行3列，即2行3列的矩阵。
二、Matrix的运算规则1. 加法和减法：两个矩阵的对应元素进行相加或相减，要求这两个矩阵的维度相同。
2. 数乘：将矩阵的每个元素与一个常数相乘。
3. 乘法：两个矩阵的乘法是指第一个矩阵的每一行与第二个矩阵的每一列进行乘积运算，并将结果相加得到新的矩阵。
要求第一个矩阵的列数等于第二个矩阵的行数。
三、Matrix在计算机科学中的应用1. 图形处理：在计算机图形学中，Matrix常用于对图像进行变换，如旋转、缩放、平移等操作。
通过矩阵运算，可以快速计算出变换后的图像的像素位置。
2. 神经网络：在机器学习和人工智能中，Matrix被广泛应用于神经网络的训练和预测过程中。
神经网络的权重和偏置通常以矩阵的形式表示，通过矩阵乘法和激活函数的组合，可以实现复杂的模式识别和预测任务。
3. 数据分析：在数据科学中，Matrix常用于处理和分析数据。
例如，通过矩阵运算可以进行数据的降维、特征提取、聚类等操作，从而揭示数据中的隐藏模式和关联关系。
4. 矩阵分解：在推荐系统中，Matrix经常被用于矩阵分解算法，通过将用户-物品关系矩阵分解为两个低维矩阵，可以实现对用户和物品的隐含特征的提取和推荐。
四、Matrix在物理学中的应用1. 量子力学：在量子力学中，Matrix被用于描述量子态和量子操作。
量子态可以通过一个复数的矩阵表示，而量子操作可以通过两个矩阵的乘法来表示。
通过矩阵运算，可以计算出量子系统的演化和测量结果。
2. 统计物理学：在统计物理学中，Matrix常用于描述系统的哈密顿量和密度矩阵。
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5、 矩阵元素的标识 矢量的生成－利用冒号(:)
a)x=i:j 步长为1,当 i<j时生成均匀等分 矢量,当i>j时,生成x空矢量; b)x=i:j:k 如： j>0,则i<k, 生成均匀等分矢量, 步长为j; 如：j<0,则i>k, 生成均匀等分矢量,步长
为-| j |;
加例:2-3 8
加例2-3 a=1:6 b=2:2:10 c=0:-1:2 %步长为-1 d=8:1:4
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1、关系运算
＜ ＜＝ ＞ 小于 小于等于 大于 ＞＝ ＝＝ ~＝ 大于等于 等于 不等于
两个标量a、b比较：成立为1，反之为0
一个 标量a和一个数组B比较：a与B中的每一个
元素进行比较，运算结果是与B同维的逻辑数组C， C由0和1组成 两个同维的数组A、B比较：A与B同位置的元素 进行比较，运算结果是与A、B同维的逻辑数组C， C由0和1组成
1
第2章 矩阵运算基础
矢量:一行或一列元素 标量:有一个元素 数组运算：以数组的元素为运算对象 矩阵运算：以矩阵整体为对象 一、 矩阵的创建
1、 命令窗口直接创建
整个矩阵元素包含在“[ ]”内
行与行之间以分号(；)或Enter键分隔；
2
行内的元素间用逗号(,)或空格分隔 ；
矩阵大小可以不必预先定义
a是矩阵，p为标量
a每一个元素与p进行幂运算：a(i).^p a是标量，p为矩阵 a与p的每一个元素进行幂运算：a.^p(i)
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【例2-11】数组a为[1 2 3]，数组b为[4 5 6]， 求数组的幂c=a.^b。 a=[1 2 3]; b=[4 5 6]; c=a.^b 【例2-12】数组a为[1 2 3]，求数组的幂c=a.^2。 a=[1 2 3]; c=a.^2 【例2-13】数组a为[1 2 3]，求数组的幂运算 c=2.^a。 a=[1 2 3]; c=2.^a
eps 相对精度 i,j 复数中的虚数单位 pi 圆周率π realmax 计算机能够显示的最大浮点数 realmin 计算机能够显示的最小浮点数 version 字符串格式的MATLAB版本
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错误提示信息说明 singular 奇异阵 Badly scaled 病态阵 inf Matlab用来表示无穷大的专用 变量，可以避免零溢出（zero overflow） Inner matrix dimensions must agree 矩阵相乘出错：维数不对
c=a' c=a.'
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三、关系运算和逻辑运算
MATLAB对此类运算符有如下规定： 所有的关系表达式或逻辑表达式中，任何非0 数都是“逻辑真”，只有0才是“逻辑假” 关系表达式或逻辑表达式的计算结果是一个 由0和1组成的“逻辑数组(Logical Array)”,数组 中1表示真，0表示假 逻辑数组是一种特殊的数值数组，对“数值 数组”操作有关命令和函数也适用它；同时又 具有自身的特殊用途 优先级别： 算术运算关系运算逻辑运算
加例：2-4
A=magic(4) A1=A(1:2,[1,2,4]) %提取第1,2行；第1,2,4列的元素 A2=A([4，1]，：) %提取第4,1行的元素，注意先后 A([1,3],[2,4])=0 %第1,3行；第2,4列的元素置零 D=A([1,3,2,1,3],[3,2,1,3,2,1]) %扩充后传递给Ｄ
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【例2-17】矩阵a和b均为2×3阶矩阵，使用 逻辑运算符计算对应元素。 a=[1 0 3;0 -1 6]; b=[-1 0 0;0 5 0.3]; a&b a|b ~a ~b
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3、关系函数与逻辑函数
any(A)判断“向量中元素是否有非0”的函数
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3、数组的乘（.*）、除法（./右除、.\左除）
数组乘、除法仍是元素间的运算，有时称 之为“点运算”。作除数的矩阵其元素应是 非零，作除数的标量也应是非零的。
同结构数组间的乘除运算
A.*B : A的元素乘以B的对应元素 A./B : A的元素除以B的对应元素 A.\B : B的元素除以A的对应元素
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【例2-16】矩阵a和b均为1×3阶矩阵，使用 关系运算符对对应元素进行比较。
a=[0 -1 2]; b=[-3 1 2]; a<b a<=b a>b a>=b a==b a~=b
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2、逻辑运算
&:逻辑与，对应元素均为非0时，结果为1，否则为0 |:逻辑或，对应元素有一个为非0时，结果为1，否则 为0 ~:“逻辑非”运算是一元运算符：非0元素的逻辑非为 “真”，用1表示；0元素的逻辑量为假，用0表示 一个标量a和一个数组B进行逻辑运算：a与B中的每 一个元素进行运算，运算结果是与B同维的逻辑数组C ，C由0和1组成 两个同结构的数组A、B逻辑运算：A与B同位置的元 素进行逻辑运算，运算结果是与A、B同维的逻辑数组 C，C由0和1组成 优先级别：逻辑非逻辑与和逻辑或同级
常用计算名词英汉对照 natural logarithm →自然对数 determinant→行列式 minus →减的,负的 norm→范数，模方 infinity →无穷大 inverse →倒数,逆 remainder →余数 pseudoinverse →伪逆 triangular →三角的 rank → 秩 conjugate →共轭 eigenvalue →特征值 exponential→指数的 eigenvector →特征向量 logarithm →对数 unitary matrix→酉矩阵 modulus →模数 orthogonal→正交的 phase →相位
矩阵元素可以是数值、表达式或函数，表 达式或函数的值是能计算出来的，不能是 未知变量 ；
矩阵元素可以是复数（complex ）:用书写 方法直接输入（虚数单位用i或j表示） ；或
用一个矩阵表示复数的实部，另一个矩阵 表示矩阵的虚部，最后将两个矩阵相加；
加例:2-1
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矩阵“[ ]”内无任何元素时，称之为空阵；
a是标量，p为方阵
a^p=v*a^d/v , [v,d]=eig(p)
a是标量，p不为方阵，则a^p是错误的
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【例2-10】矩阵a为[1 2;3 4]，求它的1.5次幂。
a=[1 2;3 4];
c=a^1.5
6、数组的幂
a.^p
a是矩阵，p为矩阵,且a与p的结构相同
a与p对应位置元素进行幂运算：a(i).^p(i)
数组B与标量c间的乘除运算
c.*B ==B.*c: B的每一个元素乘以c c./B==B.\c : c除以 B的每一个元素 c.\B ==B./c: B的每一个元素除以c
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【例2-7】两个数组相乘，数组a为 ，数组b 为 ，求两数组的乘法。
a=[1 2 3]; c1=a.*b C2=d.*a b=[4 5 6]; d=5
矩阵元素的标识:元素和子矩阵可以用标 量、矢量和冒号的标识来引用和赋值
子矩阵的序号矢量标识方式A(u,v) u,v是可以任意排列的矢量（不是特别要求， 最好使用单调序号），分别表示子矩阵元素在 母矩阵中的行号和列号。
A(u,v)的行数或列数不受限制
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“纯”冒号，形如A(L1,:)：表示子矩阵是 母矩阵L1行的所有元素， A(:,L2)：表示子 矩阵是母矩阵L2列的所有元素，A(:,:):矩阵 的所有元素 。
A(1) 1 A( 2) 4 A(3) 7 A( 4) 2 A(5) 5 A(6) 8 A(7) 3 A(8) 6 A(9) 9
【例2-9】数组a为 ，数组b为 ，求两数组的 除法。
a=[1 2 3]; b=[4 5 6]; d=5 c=a.\b c=b./a c3=d./a c4=d.\a
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4、矩阵的除法
/ 、\
矩阵除法运算要符合线性代数矩阵运算的要求：
右除a/b: a矩阵列数等于b矩阵列数
左除a\b: a矩行数等于b矩阵行数
加例2-1
A=[2 3 4 5;3 4 5 6;4 5 6 7;7 8 9 10] x=2 ; y=pi/3 A=[2,3,cos(y);x,2*x,3*x;x,x/2,x/4] %表达式 A1=[2+3i,3+4i,4;1+2i,2+3i,5;7,9,2+3i]%复数 等价： B=[2,3,4;1,2,5;7,9,2]; C=[3,4,0;2,3,0;0,0,3]; A2=B+C*i D=[]; %[]可以赋给矩阵的某一块，使之为空
作除数的矩阵应是非奇异的 【例2-8】两个矩阵相除,矩阵a和b均为3×3阶矩阵。 a=rand(3) b=rand(3) c=a/b d=b\a
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5、矩阵的幂
a是方阵
a^p
a是矩阵，p为标量
p是大于1的整数时，则a自乘p次
当p为负整数时，表示方阵直接自乘p次后的逆 当p为0时，将给出和方阵A同维的单位阵。 p为小数时，a^p=v*d.^p/v, [v,d]=eig(a)
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4、从外部数据文件调入矩阵
命令调入文本文件 格式：load filename.dat/txt 菜 单 :file→import data→ 选 择 文 件 →finish→矩阵进入工作环境 说明 :1、 文本文件中数据要排成一个矩形表,数据
之间用空格分开,一行数据即为矩阵的一行,且各行 数据元素个数必须相等 2、 文本文件中数据之间用制表符分开,形如 EXCEL的数据格式（Tab） 例：读入默认格式：laod test.mat 导入文本文件：load test.txt
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二 、 矩阵和数组的算术运算
1、矩阵和数组的加减运算