※ 引述《koei888 (^^)》之銘言：
: 1.令 A=(1-alph)I-alph*i*i^t 其中, 1>alph>0 求特性根、特性向量及逆矩陣
: A是 (n*n) 矩陣， i是單位向量 i^t 轉置單位向量
看不懂alph*i*i^t是啥(．＿．?)
: 2. show that if A is symmetirc posititve definite there exist a nonsingular
: matrix P such that P*A*P^t=I and P^t*P=A^-1
: 感謝大大
這其實是等價的 你的問題是(1)
(1) Let A be a Hermitian matrix
if A is positive definite
then there exists a nonsingular matrix P s.t. A = P P^*
(2) For any nonsingular matrix P, P P^* is Hermitian and positive definite.
pf:(1)
Since A>0, there exists a unitary matrix U s.t. U^* A U = D
where D = (d_ij) , d_ii = λ_i > 0
d_ij = 0 , i=/=j
Then A = U D U^* ---(●)
Let S = (s_ij) , s_ii = √λ_i
s_ij = 0 , i=/=j
Then D = S S = S S^* (Since S = S^*)
Finally from (●), A = U S S^* U^* = U S (U S)^* := P P^*, where P=U S
對了，是P P^*還是P^* P 隨你高興，反正P = (P^*)^*
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