线性代数第2版课件-标准正交基和正交矩阵
合集下载
相关主题
- 1、下载文档前请自行甄别文档内容的完整性，平台不提供额外的编辑、内容补充、找答案等附加服务。
- 2、"仅部分预览"的文档,不可在线预览部分如存在完整性等问题,可反馈申请退款(可完整预览的文档不适用该条件!)。
- 3、如文档侵犯您的权益，请联系客服反馈,我们会尽快为您处理(人工客服工作时间：9:00-18:30)。
1 1 0 x1 1
2
3
)
x2 x3
1 0
0 1
1
1
x2 x3
5 2
x1
4, x2
3, x3
1
1
x
1, 2 , 3
4, 3,1.
x (x, 1)1 (x, 2 )2 ( x, 3)3.
(
x,
1
)
(1,
5,
2)
1 0
6
Rn , E : e1,e2 , ,en Rn
A: a , b
?
A :1, ,i , ,r
1, ,i , ,r , 11 ii rr 0,
i
(i ,11 ii rr ) (i ,0),
1(i ,1) i (i ,i ) r (i ,r ) 0
i (i ,i ) 0, i 0 (i ,i ) | i |2 0,
01
(
)
02
1
1 0 0
1
0 0
,
2
1 0
,
3
0 1
R3
x (1,5, 2)T ,
x 1 1 52 (2) 3,
x
1, 2 , 3
1,5, 2.
x ( x,1)1 ( x,2 )2 ( x,3 )3.
1
5
(2)
A :1,2 , ,m
n
A
2
A: a (= 0) , b (≠ 0),
a1 a2
2
x1
cos
2
sin
2
B
01
(
)
02
0 0
,
2
1 0
,
3
0 1
R3
2
1
2
2 2
,
2
2 0
2 2 2
,3
2 0
0 0
1
R3
3
2 1
3 3 2
1
1
2
5
1
2
2 2
,
2
2 0
2 2 2
,3
2 0
0 0 1
R3
x (1,5, 2)T 1,2 ,3
x1, x2 , x3 .
x1
x x1 1 x22 x33 ( 1
2
3
)
x2 x3
2
2 2
2 0
2 2
2
2 0
0
0 1
x1
x2 x3
=
1 5 2
x1 3 2, x2 2 2, x3 2
x 1,2 ,3
3 2,2 2, 2
1 x (1,5, 2)T
1
2
2 2
,2
2 0
2 2 2
,3
2 0
0 0
1
3 2,2 2, 2,
a2 a1
90
B
1 0
0 1
B
| A | 1
B
1 0
0 1
a1 a2
a1 a2
x2
| B | 1
x2
A
a1
B
a2
a1 a2
a1 a2
a1
x1
8
A
cos sin
sin cos
| A | 1
A
B
cos sin
sin cos
| B | 1
x2
cos
2
sin2
AATAT ( AT )=11AT EA 2 = 1 | A | 1
7
1 1 2 1 3
1
9
8 9
4 9
A=
1 2 13
1 12
12 1
,
B=
8 9
1 9
4 9
.
4 9
4 9
7 9
A
1
12
1 2
2
1 3
2
1,
A
2
B
1
Βιβλιοθήκη Baidu
0
B
8
A
0 1
1 0
A
A
0 1
1 0
a1 a2
,
Rn ,
e1,e2 , , en
.
1e1 2e2 nen
( ,ei ) (1e1 2e2 nen ,ei ) i (ei ,ei ) i
( ,e1)e1 ( ,e2 )e2 ( ,en )en .
E : e1,e2 , ,en
( ,e1),( ,e2 ), ,( ,en ).
A A
A
AT A E.
A 1,2 , ,n ,
1T
AT A
2T
1,2 ,
T n
1T1
,n
T
21
T
n1
1T2 2T2
nT2
iT j= i , j
1,
0,
i j i j
(i ( i , j
)
1Tn
2Tn
E.
nTn
)
(i, j 1, 2, n)
A A
A •A •A • AT A E; • A1 AT . 1
i 0 , i 1, 2, , r. 1 2 r 0 1,2 , ,r
e1,e2 , , er e1,e2 , , er
e1,e2 , , er
3 Rn
V Rn
V
1 (1,0, ,0)T , 2 (0,1, ,0)T , n (0,0, ,1)T .
4
R3
1 0 0
1
1
0
(x,2) 2 2
( x,3) 1
5
2 0 = 2
1
x ( x,1)1 ( x,2 )2 ( x,3 )3.
6
1 1 0
1
1, 0
2
0, 1
3
1 1
R3
x (1,5, 2)T
1, 2 , 3
x1, x2 , x3 .
x1
x x1 1 x22 x33 ( 1
x 3 21 2 22 23
( x,1) 1
5
2
2 2 2
=3
2
2 0
( x,2 ) 1
5
2
2
2 2
=
2
2
2 0
1 x (1,5, 2)T
1
2
2 2
,2
2 0
2 2 2
,3
2 0
0 0
1
3 2,2 2, 2,
x 3 21 2 22 23
( x,1) 3 2
2
3
)
x2 x3
1 0
0 1
1
1
x2 x3
5 2
x1
4, x2
3, x3
1
1
x
1, 2 , 3
4, 3,1.
x (x, 1)1 (x, 2 )2 ( x, 3)3.
(
x,
1
)
(1,
5,
2)
1 0
6
Rn , E : e1,e2 , ,en Rn
A: a , b
?
A :1, ,i , ,r
1, ,i , ,r , 11 ii rr 0,
i
(i ,11 ii rr ) (i ,0),
1(i ,1) i (i ,i ) r (i ,r ) 0
i (i ,i ) 0, i 0 (i ,i ) | i |2 0,
01
(
)
02
1
1 0 0
1
0 0
,
2
1 0
,
3
0 1
R3
x (1,5, 2)T ,
x 1 1 52 (2) 3,
x
1, 2 , 3
1,5, 2.
x ( x,1)1 ( x,2 )2 ( x,3 )3.
1
5
(2)
A :1,2 , ,m
n
A
2
A: a (= 0) , b (≠ 0),
a1 a2
2
x1
cos
2
sin
2
B
01
(
)
02
0 0
,
2
1 0
,
3
0 1
R3
2
1
2
2 2
,
2
2 0
2 2 2
,3
2 0
0 0
1
R3
3
2 1
3 3 2
1
1
2
5
1
2
2 2
,
2
2 0
2 2 2
,3
2 0
0 0 1
R3
x (1,5, 2)T 1,2 ,3
x1, x2 , x3 .
x1
x x1 1 x22 x33 ( 1
2
3
)
x2 x3
2
2 2
2 0
2 2
2
2 0
0
0 1
x1
x2 x3
=
1 5 2
x1 3 2, x2 2 2, x3 2
x 1,2 ,3
3 2,2 2, 2
1 x (1,5, 2)T
1
2
2 2
,2
2 0
2 2 2
,3
2 0
0 0
1
3 2,2 2, 2,
a2 a1
90
B
1 0
0 1
B
| A | 1
B
1 0
0 1
a1 a2
a1 a2
x2
| B | 1
x2
A
a1
B
a2
a1 a2
a1 a2
a1
x1
8
A
cos sin
sin cos
| A | 1
A
B
cos sin
sin cos
| B | 1
x2
cos
2
sin2
AATAT ( AT )=11AT EA 2 = 1 | A | 1
7
1 1 2 1 3
1
9
8 9
4 9
A=
1 2 13
1 12
12 1
,
B=
8 9
1 9
4 9
.
4 9
4 9
7 9
A
1
12
1 2
2
1 3
2
1,
A
2
B
1
Βιβλιοθήκη Baidu
0
B
8
A
0 1
1 0
A
A
0 1
1 0
a1 a2
,
Rn ,
e1,e2 , , en
.
1e1 2e2 nen
( ,ei ) (1e1 2e2 nen ,ei ) i (ei ,ei ) i
( ,e1)e1 ( ,e2 )e2 ( ,en )en .
E : e1,e2 , ,en
( ,e1),( ,e2 ), ,( ,en ).
A A
A
AT A E.
A 1,2 , ,n ,
1T
AT A
2T
1,2 ,
T n
1T1
,n
T
21
T
n1
1T2 2T2
nT2
iT j= i , j
1,
0,
i j i j
(i ( i , j
)
1Tn
2Tn
E.
nTn
)
(i, j 1, 2, n)
A A
A •A •A • AT A E; • A1 AT . 1
i 0 , i 1, 2, , r. 1 2 r 0 1,2 , ,r
e1,e2 , , er e1,e2 , , er
e1,e2 , , er
3 Rn
V Rn
V
1 (1,0, ,0)T , 2 (0,1, ,0)T , n (0,0, ,1)T .
4
R3
1 0 0
1
1
0
(x,2) 2 2
( x,3) 1
5
2 0 = 2
1
x ( x,1)1 ( x,2 )2 ( x,3 )3.
6
1 1 0
1
1, 0
2
0, 1
3
1 1
R3
x (1,5, 2)T
1, 2 , 3
x1, x2 , x3 .
x1
x x1 1 x22 x33 ( 1
x 3 21 2 22 23
( x,1) 1
5
2
2 2 2
=3
2
2 0
( x,2 ) 1
5
2
2
2 2
=
2
2
2 0
1 x (1,5, 2)T
1
2
2 2
,2
2 0
2 2 2
,3
2 0
0 0
1
3 2,2 2, 2,
x 3 21 2 22 23
( x,1) 3 2