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首页Matrix Analysis for Scientists and Engineers工程矩阵答案整理
Matrix Analysis for Scientists and Engineers工程矩阵答案整理
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更新于2023-03-16
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课本Matrix Analysis for Scientists and Engineers的答案整理
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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:
(a)
2
2
2
2
2cos sin 2
1
4
sin2 2sin
A
4 2 2 2
2 2 4 2
4cos sin 2 2(cos sin )sin2
1
4
2(cos sin )sin2 4sin sin 2
42
42
4cos (2sin cos ) 2sin2
1
4
2sin2 4sin (2sin cos )
2 2 2
2 2 2
4cos (sin cos ) 2sin2
1
4
2sin2 4sin (sin cos )
2
2
2cos sin2
1
2
sin2 2sin
A
Therefore, A is idempotent for all
.
2-3 Let
12
,
n
v v v
be orthonomal vectors in
n
R
. Show that
n
AvAvAv
21
,
are also orthonomal if and only if
nn
RA
is orthogonal.
proof: 1)
n
AvAvAv
21
,
are orthonomal →
nn
RA
is orthogonal
Assumed that B={
n
AvAvAv
21
,
},then B is orthogonal
1 i=j
as
n
vvv
21
,
are orthonomal vectors in
n
R
,then
j
T
i
vv
0
otherwise
So
TTT
ii
n
i
TT
nAAAvAvBBBB
1
as B is orthogonal, so as A
2)
nn
RA
is orthogonal→
n
AvAvAv
21
,
are orthonomal
If A is orthogonal then
IAAAA
TT
,
1 i=j
j
T
ij
TT
ij
T
i
vvAvAvAvAv )()(
0 otherwise
So
n
AvAvAv
21
,
are orthonomal vectors.
2.6. Prove Theorem 2.22(for the case of two subspaces R and S only).
1)
SV
(in general ,
1k
1
+ + =:
k
i
i
V
,for finite k).
2)
SV
(in general ,
A
V
for an arbitrary index set A).
Prove: 1).
{ : , }, ,S r s r s S for S V
12
, RS
,
12
,K K F
111
rs
,
222
rs
1 1 2 2 1 1 2 2 1 1 2 2
( ) ( )K K K r K r K s K s
1 1 2 2
K r K r R
,
1 1 2 2
K s K s R
1 1 2 2
K K R S
2)
{ : }S V v andv S
12
, RS
,
12
,K K F
1
R
,
1
S
2
R
,
2
S
1 1 2 2
K K R
1 1 2 2
K K S
Then ,
SV
2.7 let
denote the vector space of polynomials of degree less than or equal to , and of
the form
,where the coefficients
are real. Let
denote
the subspace of all even polynomials in
, i.e., those that satisfy the property
. Similarly, let
denote the subspace of all odd polynomials, i.e.,
.
Show that
.
Proof:
a) proof
:
From the above we can see
Only
in the set
;
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