每週問題 October 1, 2012

本週問題是利用奇異值分解證明 A^{\ast}A 么正相似 (unitarily similar) 於 AA^{\ast}

Let A be an n\times n matrix. Show that A^\ast A and AA^\ast are unitarily similar, i.e., there exists a unitary matrix Q such that

AA^\ast=QA^\ast AQ^\ast.

Note that Q is unitary if QQ^\ast = Q^\ast Q = I.

 
參考解答:

A 的奇異值分解為 A = U\Sigma V^\ast,其中 \Sigma 為一 n\times n 階實對角矩陣,UVn\times n 階么正 (unitary) 矩陣。計算

A^\ast A = V\Sigma U^\ast =U\Sigma V^\ast= V\Sigma^2V^\ast

AA^\ast= U\Sigma V^\ast V\Sigma U^\ast= U\Sigma^2 U^\ast=(UV^\ast)(V\Sigma^2V^\ast)VU^\ast=Q(A^\ast A)Q^\ast

其中 Q = UV^\ast 是么正矩陣,因為 QQ^\ast=UV^\ast VU^\ast = UU^\ast = I

PowSol-Oct-1-12

This entry was posted in pow 二次型, 每週問題 and tagged , , . Bookmark the permalink.

Leave a Reply

Fill in your details below or click an icon to log in:

WordPress.com Logo

You are commenting using your WordPress.com account. Log Out /  Change )

Google photo

You are commenting using your Google account. Log Out /  Change )

Twitter picture

You are commenting using your Twitter account. Log Out /  Change )

Facebook photo

You are commenting using your Facebook account. Log Out /  Change )

Connecting to %s