每週問題 May 29, 2017

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)^\perpN(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

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