矩阵分析复习题 en

18. Define it in detail.
x ≡ (| 2 x1 − x2 |2 + | 3 x2 |2 )1 2 , is this a vector norm on C 2 ? Explain
19. Given A( x) = ⎜ ⎜
⎛1 ⎝x
x2 ⎞ ⎟， 1⎟ ⎠
d 3 A( x) dA −1 ( x) Compute and , dx dx 3
（2）Find the range and the kernel of f.
⎛17 0 − 25 ⎞ ⎜ ⎟ A=⎜ 0 1 0 ⎟ ⎜ 9 0 − 13 ⎟ ⎝ ⎠
T
2. Given
（1）Find the Jordan canonical form and the corresponding similarity matrix of A. （2）compute e A , e At .
2 0⎤ ⎥ 0 1⎥ . 0 0⎥ ⎦
⎛0 −1 i ⎞ ⎜ ⎟ 12. Find the spectral decomposition of A = ⎜ 1 0 0 ⎟ . ⎜ i 0 0⎟ ⎝ ⎠ 0 ⎞ ⎛1 / 6 0 ⎜ ⎟ 13. Given A = ⎜ 0 0.5 1 ⎟ ⎜ 0 0 0 .5 ⎟ ⎝ ⎠
T
⎡ 1 0 −1⎤ ⎥ A= ⎢ ⎢1 1 0⎥ ⎢ ⎣ −1 2 3 ⎥ ⎦ （ 1 ） Find the matrix representation of
f with respect to the basis
β1 = [3, 0, 0]T , β 2 = [0, 2, 0]T , β3 = [0, 0,1] ;
n× n
.
17. Let A ∈ C m×n ，show that the Frobenius norm A
F
= (Tr( AH A)) 2
1
is unitarily invariant ， that is, for any unitary matrices U and V , we have A F = UAV F .
Show that： ∑ k Ak −1 converges. And compute ∑ k Ak −1 .
n =1 n =1
∞
∞
14.
0 ⎤ ⎡1 0 ⎢ k k = 1, 2,… , and lim Ak . A = ⎢0 0.9 0 ⎥ ⎥ ， compute A , k →∞ ⎢ ⎥ 0 1 0.9 ⎣ ⎦
3. Compute e A and sin A ，where
π A= ⎢ 0 1 0⎥ ⎢ ⎥. 2
⎢ ⎣0 0 2⎥ ⎦
⎡1 1 0 ⎤
4. Let A, B ∈ C n ，Show that det(e
ຫໍສະໝຸດ Baidu
A+ B
) = det(e A )det(e B ) .
5. For any A ∈ C n×n ，show that： e 2π iI = I , e 2π iI + A = e A . 6. (1) (2) (3) Let A be a normal matrix, then A is Hermitian if and only if each eigenvalue of A is real. A is shew-Hermitian if and only if each eigenvalue of A has real part equal to 0. A is unitary if and only if each eigenvalue of A has absolute value 1.
15.
Let
A be
1 2
a
positive
definite
Hermitian
matrix ， show
that ：
x = ( x H Ax) , 16. Show that：
∀x ∈ C n , is a vector norm on C n
A
n max aij , ∀A ∈ C n×n
i, j
is a matrix norm on C
⎛1 2⎞ ⎜ ⎟ 10. Given A = ⎜ 0 0 ⎟ ⎜0 0⎟ ⎝ ⎠
（1） Find the SVD of A. （2） Find the Pseudo-inverse of A.
⎡ 2 ⎢ 11. Find the polar decomposition of A = ⎢ 0 ⎢ 0 ⎣
∫
A( x )dx , ( ∫ A( x )dx )' . 0 0
1
x3
20. Given the following inconsistent linear equations
⎧ x1 + 2 x2 + 3 x3 = 1 ⎪ x1 + x3 = 0 ⎪ ⎨ ⎪ 2 x1 + 2 x3 = 1 ⎪ ⎩2 x1 + 4 x2 + 6 x3 = 1 Find the best least –squares solution.
7. Let A be a positive semi-definite Hermitian matrix, and A ≠ 0, show that： (1) det( A + E ) > 1.
⎛ ∞ Ak ⎞ (2) det ⎜ ∑ ⎟ > 1. ⎝ k =0 k ! ⎠
8 . Given a complex quadratic form： f ( x1 , x2 , x3 ) = x1 x1 + 3ix1 x3 + 4 x2 x2 − 3ix3 x1 + x3 x3 , Find a unitary transformation X=UY such that f ( x1 , x2 , x3 ) is a standard form. 9. Let A be a positive semi-definite Hermitian matrix, and A ≠ 0. B is a positive definite Hermitian matrix. Prove that : det (A+B)>det (B).
1.Let f be a linear transformation onC 3 .The matrix representation of f is A with respect to the basis α1 = [1,1,1]T , α 2 = [1,1, 0]T , α 3 = [1, 0, 0] .