每週問題 May 29, 2017

Posted on 05/29/2017 by ccjou

這是零空間的包容關係與矩陣乘法的問題。

Let A and B be complex matrices of size m\times n and p\times n, respectively. If N(A)\subset N(B), prove that B=XA for some p\times m matrix X.

 
參考解答：

底下以 C(M) 表示矩陣 M 的行空間 (column space)。使用正交補集性質 N(A)=C(A^\ast)^\perp 與 N(B)=C(B^\ast)^\perp，推得

\begin{aligned} N(A)\subset N(B)&\Rightarrow C(A^\ast)^\perp\subset C(B^\ast)^\perp\\ &\Rightarrow C(B^\ast)\subset C(A^\ast) \end{aligned}

因此，B^\ast 的每個行可表示為 A^\ast 的行向量的線性組合，也就是存在 m\times p 階矩陣 P 使得 B^\ast=A^\ast P，取共軛轉置，B=P^\ast A。

Advertisement
This entry was posted in pow 內積空間, 每週問題 and tagged . Bookmark the permalink.

Leave a Reply

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

Gravatar
WordPress.com Logo

You are commenting using your WordPress.com 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 )

Cancel

Connecting to %s