一组非奇异H-矩阵的新判据

㊀㊀第39卷㊀第3期㊀吉首大学学报(自然科学版)V o l .39㊀N o .3㊀㊀㊀㊀2018年5月J o u r n a l o f J i s h o uU n i v e r s i t y (N a t u r a l S c i e n c eE d i t i o n )M a y 2018㊀㊀文章编号:10072985(2018)03000905一组非奇异H 矩阵的新判据∗张林娟,莫宏敏(吉首大学数学与统计学院,湖南吉首416000)㊀㊀摘㊀要:利用α对角占优矩阵的元素性质和不等式性质,通过添加适当的参数,给出了一组非奇异H 矩阵的判定条件,并用数值实例验证了判定方法的有效性.关键词:非奇异H 矩阵;α对角占优矩阵;不等式;判定中图分类号:O 151.21㊀㊀㊀㊀㊀㊀㊀文献标志码:AD O I :10.3969/j .c n k i .jd x b .2018.03.003H 矩阵是特殊矩阵中最重要的一类矩阵,在计算数学㊁经济数学㊁物理学电力系统理论和控制论中都有广泛的应用.笔者拟在文献[14]的基础上给出一组判定非奇异H 矩阵的方法.1㊀相关定义设C n ˑn (R n ˑn )表示所有的n ˑn 阶复(实)数矩阵的集合,A =(a i j )n ˑn ɪC n ˑn,a i i ʂ0,i ɪN ={1,2, ,n }.记R i (A )=ðj ʂi |a i j |,S j (A )=ði ʂj|a i j |,i ,j ɪN .定义1[5]㊀设A =(a i j )n ˑn ɪC n ˑn,若对于∀i ɪN ,有|a i i |>R i (A ),则称A 为严格的对角占优矩阵;若存在正对角矩阵D 使得A D 为严格的对角占优矩阵,则称A 为非奇异H 矩阵,记为A ɪʏD .定义2[5]㊀设A =(a i j )n ˑn ɪC n ˑn,若存在αɪ(0,1],对于∀i ɪN ,有|a i i |>αR i (A )+(1-α)S i (A ),则称A 为严格的α对角占优矩阵;若存在正对角矩阵D 使得A D 为严格的α对角占优矩阵,则称A 为广义的α对角占优矩阵.为了叙述方便,引入下列划分:N 1={i ɪN :0<|a i i |=αR i (A )+(1-α)S i (A )},N 2={i ɪN :0<|a i i |<αR i (A )+(1-α)S i (A )},N 3={i ɪN :|a i i |>αR i (A )+(1-α)S i (A )}.显然,N =N 1ʏN 2ʏN 3.定义r =m a x r i =αðt ɪN 1|a i t |+ðt ɪN 2|a i t |()+(1-α)S i (A )|a i i |-αðt ɪN 3,t ʂi|a i t |æèöø㊀㊀i ɪN 3,∗收稿日期:20171225基金项目:吉首大学校级科研项目(J D Y 16002)通信作者:莫宏敏(1969 ),男(土家族),湖南慈利人,吉首大学数学与统计学院副教授,博士,主要从事矩阵理论与计算研究.δi (A )=αðt ɪN 1|a i t |+ðt ɪN 2|a i t |+r ðt ɪN 3|a i t |()+(1-α)S i (A )|a i i |.2㊀主要结果及其证明引理1[6]㊀设A =(a i j )ɪC n ˑn,若A 为广义的α对角占优矩阵,则A 为非奇异H 矩阵.引理2[7]㊀设A =(a i j )ɪC n ˑn,若A 为不可约的α对角占优矩阵,或A 为具有非零元素链的α对角占优矩阵,则A 为非奇异H 矩阵.定理1㊀设A =(a i j )ɪC n ˑn ,若存在αɪ0,1(],对于∀i ɪN 2,有|a i i |>δi (A )αδi (A )-1ðt ɪN 1|a i t |+ðt ɪN 2,t ʂi 1-1δt (A )æèöø|a i t |+ðt ɪN3δt (A )|a i t |æèöø+(1-α)S i (A ),(1)则A 为非奇异H 矩阵.证明㊀由r 和δi (A )的定义可知δi (A )ɤr ,i ɪN 3且0ɤr <1.对于∀i ɪN 1,记P i (A )=αðt ɪN 1,t ʂi |a i t |+ðt ɪN 21-1δt (A )æèöø|a i t |+ðt ɪN 3δt (A )|a i t |æèöø+(1-α)S i (A );对于∀i ɪN 2,记Q i (A )=αðt ɪN 1|a i t |+ðt ɪN 2,t ʂi 1-1δt (A )æèöø|a i t |+ðt ɪN 3δt (A )|a i t |æèöø+(1-α)δi (A )-1δi (A )S i (A ).根据N 1,N 2,N 3的划分,易知对于∀t ɪN 2,有0<δt (A )-1δt (A )<1,对于∀t ɪN 3,有0<δt <1,于是αR i (A )+(1-α)S i (A )>P i (A )㊀㊀i ɪN 1.(2)设d i =1αðt ɪN 3|a i t |(|a i i |-P i (A ))㊀㊀∀i ɪN 1,1αðt ɪN 3|a i t |δi (A )-1δi (A )|a i i |-Q i (A )æèöø㊀㊀∀i ɪN 2.ìîí(3)当αðt ɪN 3|a i t |=0时,记d i =+ɕ.因为根据(1),(2)式可知,对于∀i ɪN 1ɣN 2,有d i >0,所以存在充分小的ε(ε>0),使得0<ε<m i n i ɪN 1ɣN 2d i ɤ+ɕ.(4)构造正对角矩阵X =d i a g(x 1,x 2, ,x n ),其中x i =1㊀㊀i ɪN 1,δi (A )-1δi (A )㊀㊀i ɪN 2,ε+δi (A )㊀㊀i ɪN 3.ìîí令B =(b i j )n ˑn =A X ,对于∀i ,j ɪN ,有b i j =x j a i j .现只要证明B 为严格的α对角占优矩阵即可.(ⅰ)对于∀i ɪN 1.当αðt ɪN 3|a i t |ʂ0时,由(2) (4)式可知|b i i |=|a i i |>αðt ɪN 1,t ʂi |a i t |+ðt ɪN 21-1δt (A )æèöø|a i t |+ðt ɪN 3(d i +δt (A ))|a i t |æèöø+(1-α)S i (A )>αðt ɪN 1,t ʂi |a i t |+ðt ɪN 21-1δt (A )æèöø|a i t |+ðt ɪN 3(ε+δt (A ))|a i t |æèöø+(1-α)S i (A )=αR i (B )+(1-α)S i (B );当αðt ɪN 3|a i t |=0时,因αɪ(0,1]ʂ0,故对于∀i ɪN 3,有|a i t |=0.由(2),(3)式可知01吉首大学学报(自然科学版)第39卷|b i i |=|a i i |=αR i (A )+(1-α)S i (A )>αðt ɪN 1,t ʂi |a i t |+ðt ɪN 21-1δt (A )æèöø|a i t |æèöø+(1-α)S i (A )=P i (A )=αR i (B )+(1-α)S i (B ).(ⅱ)对于∀i ɪN 2.当αðt ɪN 3|a i t |ʂ0时,由(2) (4)式可知δi (A )-1δi (A )|a i i |=αðt ɪN 1|a i t |+ðt ɪN 2,t ʂi 1-1δt (A )æèöø|a i t |+ðt ɪN 3(d i +δt (A ))|a i t |æèöø+(1-α)δi (A )-1δi (A )S i (A )>αðt ɪN 1|a i t |+ðt ɪN 2,t ʂi 1-1δt (A )æèöø|a i t |+æè㊀㊀ðt ɪN 3(ε+δt (A ))|a i t |öø+(1-α)δi (A )-1δi (A )S i (A )=αR i (B )+(1-α)S i (B );当αðt ɪN 3|a i t |=0时,因αɪ(0,1]ʂ0,故对于∀i ɪN 3,有|a i t |=0.由(1)式可知|b i i |=δi (A )-1δi (A )|a i i |>αðt ɪN 1|a i t |+ðt ɪN 21-1δt (A )æèöø|a i t |+ðt ɪN 3,t ʂiδt (A )|a i t |æèöø+(1-α)δi (A )-1δi (A )S i (A )=αðt ɪN 1|a i t |+ðt ɪN 21-1δt (A )æèöø|a i t |æèöø+(1-α)δi (A )-1δi(A )S i (A )=αR i (B )+(1-α)S i (B ).(ⅲ)对于∀i ɪN 3.由定义式可知:对于∀t ɪN 2,有δt (A )>1且0<δt (A )-1δt (A )<1;对于∀t ɪN 3,有0<δt (A )<1.故对于∀i ɪN 3,有R i (A )-ðt ɪN 1|a i t |+ðt ɪN 21-1δt (A )æèöø|a i t |+ðt ɪN 3,t ʂiδt (A )|a i t |æèöø>0.又由N 3的定义可知,对于∀i ɪN 3,有|a i i |-αðt ɪN 3,t ʂi|a i t |+(1-α)S i (A )()>0.于是对于∀i ɪN 3,有|b i i |-(αR i (B )+(1-α)S i (B ))=(ε+δi (A ))|a i i |-αðt ɪN 1,t ʂi |a i t |+ðt ɪN 2,t ʂi 1-1δt (A )æèöø|a i t |+æèæè㊀㊀ðt ɪN 3,t ʂi (ε+δt (A ))|a i t |öø+(1-α)(ε+δi (A ))S i (A )öø=ε㊀㊀|a i i |-㊀㊀αðt ɪN 3,t ʂi|a i t |+æèæè㊀㊀(1-α)S i (A )öøöø+δi (A )|a i i |-αðt ɪN 1|a i t |+ðt ɪN 21-1δt (A )æèöø|a i t |+æèæè㊀㊀ðt ɪN 3,t ʂi δt (A )|a i t |öø+(1-α)δi (A )S i (A )öø>α㊀㊀R i (A )-㊀㊀ðt ɪN 1|a i t |+æèæè㊀㊀ðt ɪN 21-1δt (A )æèöø|a i t |+㊀㊀ðt ɪN 3,t ʂiδt (A )|a i t |öøöø+(1-α)S i (A )-(1-α)δi (A )S i (A )ȡ0,即|b i i |>αR i (B )+(1-α)S i (B ).综合(ⅰ) (ⅲ)可知,对于∀i ɪN ,有|b i i |>αR i (B )+(1-α)S i (B ),即B 为严格的α对角占优矩阵.由B =A X 可知A 为广义的α对角占优矩阵,再由引理1可知A 为非奇异H 矩阵.定理2㊀设A =(a i j )ɪC n ˑn,A 不可约,若存在αɪ0,1(],对于∀i ɪN 2,有|a i i |ȡδi (A )αδi (A )-1ðt ɪN 1|a i t |+ðt ɪN 2,t ʂi 1-1δt (A )æèöø|a i t |+ðt ɪN3δt (A )|a i t |æèöø+(1-α)S i (A ),(5)11第3期㊀㊀㊀㊀㊀㊀㊀㊀㊀㊀㊀㊀㊀㊀㊀㊀张林娟,等:一组非奇异H 矩阵的新判据且至少有1个不等式是严格成立的,则A 为非奇异H 矩阵.证明㊀因为A 不可约,所以存在正对角矩阵X 使得A X 不可约.令B =(b i j )n ˑn =A X ,b i j =x j a i j ,∀i ,j ɪN ,则B 也不可约.下面证明B 为不可约的α对角占优矩阵.因为对于∀i ɪN 2,(5)式中至少有1个不等式严格成立,不妨设j ɪJ k ⊂N 2,所以|a j j |>δj (A )αδj (A )-1ðt ɪN 1|a j t |+ðt ɪN 2,t ʂj 1-1δt (A )æèöø|a j t |+ðt ɪN 3δt (A )|a j t |æèöø+(1-α)S j (A ).构造正对角矩阵X =d i a g(x 1,x 2, ,x n ),其中x i =1㊀㊀i ɪN 1,δi (A )-1δi (A )㊀㊀i ɪN 2,δi (A )㊀㊀i ɪN 3.ìîí(ⅰ)对于∀i ɪN 1,有|b i i |>αR i (B )+(1-α)S i (B ).(ⅱ)对于∀i ɪN 2\J k ,有|b i i |=δi (A )-1δi (A )|a i i |=αðt ɪN 1|a i t |+ðt ɪN 2,t ʂi 1-1δt (A )æèöø|a i t |+ðt ɪN3δt (A )|a i t |æèöø+(1-α)δi (A )-1δi (A )S i (A )=αR i (B )+(1-α)S i (B );对于∀i ɪJ k ⊂N 2,有|b i i |=δi (A )-1δi (A )|a i i |>αðt ɪN 1|a i t |+ðt ɪN 2,t ʂi 1-1δt (A )æèöø|a i t |+ðt ɪN 3δt (A )|a i t |æèöø+(1-α)δi (A )-1δi (A )S i (A )=αR i (B )+(1-α)S i (B ).(ⅲ)对于∀i ɪN 3,有|b i i |-(αR i (B )+(1-α)S i (B ))=δi (A )|a i i |-α㊀㊀ðt ɪN 1|a i t |+ðt ɪN 2,t ʂi 1-1δt (A )æèöø|a i t |+æèæè㊀㊀ðt ɪN 3δt (A )|a i t |öø+(1-α)δi (A )S i (A )öø=αR i (A )+(1-α)S i (A )-α㊀㊀ðt ɪN 1|a i t |+ðt ɪN 2,t ʂi 1-1δt (A )æèöø|a i t |+ðt ɪN 3δt (A )|a i t |æèöø-(1-α)δi (A )S i (A )>(1-α)(1-δi (A ))S i (A )ȡ0.综合(ⅰ) (ⅲ)可知,对于∀i ɪN ,有|b i i |ȡαR i (B )+(1-α)S i (B ),B 不可约且至少有1个不等式严格成立,即B 为不可约的α对角占优矩阵.由引理2可知B 为非奇异H 矩阵,再由引理1可知A 为非奇异H 矩阵.定理3㊀设A =(a i j )ɪC n ˑn,若存在αɪ0,1(],满足以下条件:(ⅰ)K (A )=i ɪN 2:|a i i |=δi (A )αδi (A )-1ðt ɪN 1|a i t |+ðt ɪN 2,t ʂi 1-1δt (A )æèöø|a i t |+æè{㊀㊀ðt ɪN 3δt (A )|a i t |öø+(1-α)S i (A )}ʂØ,且对于∀i ɪK (A ),存在非零元素链a i j 1a j 1j 2 a j k -1j k使得j k ɪN \K (A );(ⅱ)对于∀i ɪN 2,有|a i i |ȡδi (A )αδi (A )-1ðt ɪN 1|a i t |+ðt ɪN 2,t ʂi 1-1δt (A )æèöø|a i t |+ðt ɪN 3δt (A )|a i t |æèöø+(1-α)S i (A ).则A 为非奇异H 矩阵.证明同定理1,略.21吉首大学学报(自然科学版)第39卷3㊀数值实例设A =5220610100802010801000421010150023130æèöø.取α=1,根据文献[1],N 的划分为N 1=Ø,N 2={1,4},N 3={2,3,5}.因a 11=5,a 22=100,a 33=100,a 44=15,a 55=30,故R 1(A )=10,R 2(A )=92,R 3(A )=94,R 4(A )=22,R 5(A )=6.于是,|a 11|=5>4.92=|a 14|+|a 12|R 2(A )|a 22|+|a 13|R 3(A )|a 33|+|a 15|R 5(A )|a 55|æèöø,|a 44|=15<20.6=|a 41|+|a 42|R 2(A )|a 22|+|a 43|R 3(A )|a 33|+|a 45|R 5(A )|a 55|æèöø,无法用文献[1]中的定理1进行判定.取α=1219,令r =1,根据文献[5],N 的划分为N 1={4},N 2={1},N 3={2,3,5}.于是,|a 11|=5ɱ5.0629=δ1(A )αδ1(A )-1|a 14|+δ2(A )|a 12|+δ3(A )|a 13|+δ5(A )|a 15|(),无法用文献[5]中的定理1进行判定.当α=1219时,根据定理1,N 的划分为N 1={4},N 2={1},N 3={2,3,5}.于是,r 2=0.8787,r 3=0.8800,r 5=0.1608,r =m a x i ɪN 3(r 2,r 3,r 5)=0.8800,δ1(A )=2.7326,δ2(A )=0.8667,δ3(A )=0.8800,δ4(A )=0.8989,δ5(A )=0.2365.从而,|a 11|=5>4.8932=δ1(A )αδ1(A )-1(|a 14|+δ2(A )|a 12|+δ3(A )|a 13|+δ5(A )|a 15|),由定理1可判定A 为非奇异H 矩阵.参考文献:[1]黄延祝.非奇H 矩阵的简捷判据[J ].计算数学,1993,15(3):318328.[2]江㊀如.广义对角占优矩阵的新判据[J ].华南师范大学学报(自然科学学版),2010(1):2427.[3]高慧敏,陆㊀全,徐㊀仲,等.非奇H 矩阵的一组含参数迭代判定准则[J ].高校应用数学学报,2012,27(4):439448.[4]刘建州,吕振华,李㊀林,等.一组非奇异H 矩阵的实用判据[J ].湖南文理学院学报(自科版),2015,27(2):34;13.[5]江㊀如.非奇异H 矩阵的新判据[J ].工程数学学报,2011,28(3):393400.[6]李继成,张文修.H 矩阵的判定[J ].高等学校计算数学学报,1999,21(3):264268.[7]孙玉祥.广义对角占优矩阵的充分条件[J ].高等学校计算数学学报,1997,19(3):216223.S o m eN e wC r i t e r i a f o rN o n s i n gu l a rH -M a t r i c e s Z H A N GL i n j u a n ,MO H o n gm i n (C o l l e g e o fM a t h e m a t i c s a n dS t a t i s t i c s ,J i s h o uU n i v e r s i t y,J i s h o u416000,H u n a nC h i n a )A b s t r a c t :B y m a k i n g u s e o f t h e e l e m e n t p r o p e r t i e s o f α-d i a g o n a l l y d o m i n a n tm a t r i x a n d a d d i n g t h e a p pr o -p r i a t e p a r a m e t e r s ,w e p r e s e n t a s e t o f d e t e r m i n a t i o nc o n d i t i o n s f o rn o n s i n g u l a rH -m a t r i xa n dv e r i f y th e e f f e c t i v e n e s s o f t h em e t h o db y n u m e r i c a l e x a m pl e s .K e y wo r d s :n o n s i n g u l aH -m a t r i x ;α-d i a g o n a l l y d o m i n a n tm a t r i x ;i n e q u a l i t y ;d e t e r m i n a t i o n (责任编辑㊀向阳洁)31第3期㊀㊀㊀㊀㊀㊀㊀㊀㊀㊀㊀㊀㊀㊀㊀㊀张林娟,等:一组非奇异H 矩阵的新判据。