Matrix Analysis for Scientists and Engineers工程矩阵答案整理
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2015-02-28
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1.1 If
nn
AR
and
is a scalar, what is det (
A
)? What is det (
A
)?
Solution:
Det (
A
) =
det( )
n
A
.
Det (
A
) =
( 1) det( )
n
A
.
1.2 If A is orthogonal, what is det A? If A is unitary, what is det A?
Solution:
(1) If A is orthogonal, det A is +1 or -1.
det( )det( ) det( )det( ) det( ) det( ) 1
det( ) 1
TT
A A A A A A I
A
(2) If A is unitary,
det( ) 1 ( )
nn
AA
det( ) det( ) det( ) det( ) det( ) det( ) det( ) det( ) 1
det( ) 1
HH
A A A A A A AA I
A
1.3 Let
,
n
x y R
. Show that
det( )
TT
I xy I y x
.
Proof:
Let
2
2
[ , ]
|| ||
x
XX
x
be an orthogonal matrix with first column
2
|| ||
x
x
.
Then
det( - ) det( ( - ) ) det( - )
T T T T T
I xy X I xy X I X xy X
.
We have
[|| ||,0,0,...,0]
TT
X x x
, and
2
2
[ , ]
|| ||
T
TT
yx
y X y X
x
.
So,
1-
-
0
TT
TT
y x y X
I X xy X
I
.
Which is upper triangular with determinate
1
T
yx
.
1.6 A matrix
nn
A
is said to be idempotent if
2
AA
.
(a) Show that the matrix
2
2
2cos sin2
1
2
sin2 2sin
A
is idempotent for all
.
(b)Suppose
nn
A
is idempotent and
AI
. Show that A must be singular.
Solution:
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weixin_421327772019-01-22下载了用edge打不开啊，应该用什么打开呢- 以后还能再见2017-10-14涵盖了不少题的答案！
- Pompeii012020-08-31乱码很多，而且一共就14页，内容几乎无用...不知道17年10月14的哥们咋给的5星。
- Kang_SF2015-10-02非常有用，但下载了却打不开文件啊
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