Linear Algebra, Theory And Applications, 2012a

13.12. EXERCISES 335
8. Prove the Cayley Hamilton theorem as follows. First suppose A has a basis of eigenvectors
{v k } n k=1 ,Av k = λ k v k . Let p (λ) be the characteristic polynomial. Show
p (A) v k = p (λ k ) v k = 0. Then since {v k } is a basis, it follows p (A) x = 0 for all
x and so p (A) =0. Next in the general case, use Problem 7 to obtain a sequence {A k }
of matrices whose entries converge to the entries of A such that A k has n distinct
eigenvalues and therefore by Theorem 7.1.7 A k has a basis of eigenvectors. Therefore,
from the first part and for p k (λ) the characteristic polynomial for A k , it follows
p k (A k )=0. Now explain why and the sense in which lim k→∞ p k (A k )=p (A) .
9. Prove that Theorem 13.4.6 and Corollary 13.4.7 can be strengthened so that the
condition ( on ) the A k is necessary as well as sufficient. Hint: Consider vectors of the
x
form where x ∈ F
0
k .
10. Show directly that if A is an n × n matrix and A = A ∗ (A is Hermitian) then all the
eigenvalues are real and eigenvectors can be assumed to be real and that eigenvectors
associated with distinct eigenvalues are orthogonal, (their inner product is zero).
11. Let v 1 , ··· , v n be an orthonormal basis for F n . Let Q be a matrix whose i th column
is v i . Show
Q ∗ Q = QQ ∗ = I.
12. Show that an n × n matrix Q is unitary if and only if it preserves distances. This
means |Qv| = |v| . This was done in the text but you should try to do it for yourself.
13. Suppose {v 1 , ··· , v n } and {w 1 , ··· , w n } are two orthonormal bases for F n and suppose
Q is an n × n matrix satisfying Qv i = w i . Then show Q is unitary. If |v| =1,
show there is a unitary transformation which maps v to e 1 .
14. Finish the proof of Theorem 13.6.5.
15. Let A be a Hermitian matrix so A = A ∗ and suppose all eigenvalues of A are larger
than δ 2 . Show
(Av, v) ≥ δ 2 |v| 2
Where here, the inner product is (v, u) ≡ ∑ n
j=1 v ju j .
16. Suppose A + A ∗ has all negative eigenvalues. Then show that the eigenvalues of A
have all negative real parts.
17. The discrete Fourier transform maps C n → C n as follows.
F (x) =z where z k = √ 1
n−1
∑
e −i 2π n jk x j .
n
Show that F −1 exists and is given by the formula
F −1 (z) =x where x j = √ 1
n−1
∑
e i 2π n jk z k
n
Here is one way to approach this problem. Note z = Ux where
⎛
⎞
e −i 2π n 0·0 e −i 2π n 1·0 e −i 2π n 2·0 ··· e −i 2π n (n−1)·0
U = √ 1
e −i 2π n 0·1 e −i 2π n 1·1 e −i 2π n 2·1 ··· e −i 2π n (n−1)·1
e
n −i 2π n 0·2 e −i 2π n 1·2 e −i 2π n 2·2 ··· e −i 2π n (n−1)·2
⎜
⎝
.
.
.
⎟
. ⎠
e −i 2π n 0·(n−1) e −i 2π n 1·(n−1) e −i 2π n 2·(n−1) ··· e −i 2π n (n−1)·(n−1)
j=0
j=0