每週問題 March 20, 2017

證明 A+B 為正規矩陣 (normal matrix) 是一個充分條件。

Let A and B be normal matrices such that C(A)\perp C(B), where C(X) denotes the column space of X. Prove that A+B is a normal matrix. Note that P is a normal matrix if P^\ast P=PP^\ast.

 
參考解答:

條件 C(A)\perp C(B) 意味 C(A)\subset C(B)^\perp=N(B^\ast),可知 B^\ast A=0。同樣道理,A^\ast B=0。因此,

\begin{aligned} (A+B)^\ast(A+B)&=A^\ast A+A^\ast B+B^\ast A+B^\ast B\\ &=A^\ast A+B^\ast B.\end{aligned}

再者,正規矩陣 AB 滿足 C(A)=C(A^\ast)C(B)=C(B^\ast),就有 C(A^\ast)\perp C(B^\ast)。因此,BA^\ast=0AB^\ast=0,我們得到

\begin{aligned} (A+B)(A+B)^\ast&=AA^\ast+AB^\ast+BA^\ast+BB^\ast\\ &=AA^\ast+BB^\ast\\ &=A^\ast A+B^\ast B.\end{aligned}

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

1 Response to 每週問題 March 20, 2017

  1. Meiyue Shao says:

    用谱分解来证也很方便.

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