Re: [線代] 兩小題特殊矩陣的性質
※ 引述《cxcxvv (delta)》之銘言：
: _T
: 1.If A is Hermitian(A=A), then A+iI is invertible.
: 2.If Q is orthogonal, then Q+0.5I is invertible.
: 做很久證明不了也舉不出反例
: 還請高手幫忙 謝謝
以第1題為例，
A+iI is invertible if and only if the nullspace of A+iI is {0}.
Assume that (A+iI)x=0 for some x \in C^n.
Then Ax = -ix.
然後用 Euclidean Inner Product 去做。
_T
用B^*表示 B。
<v,w> = v^* w
(Ax)^* Ax = (-ix)^* (-ix) = x^* x
||
x^* (A^* A) x
= x^* A^2 x
= x^* A(-ix)
= x^* - x = -(x^* x)
=> x^* x = 0 => x=0.
用一樣的思維也可以解決第二題。
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◆ From: 182.235.189.13
※ 編輯: Eeon 來自: 182.235.189.13 (12/18 04:36)
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