每週問題 May 23, 2016

這是關於矩陣秩的擾動問題。

Let A and B be m\times n matrices. If \hbox{rank}A=r and \hbox{rank}B=k\le r, show that

r-k\le \hbox{rank}(A+B)\le r+k.

In words, a perturbation of rank k can change the rank by at most k.

 
參考解答:

使用不等式 \hbox{rank}(A+B)\le\hbox{rank}A+\hbox{rank}B,即 \hbox{rank}(A+B)\le r+k。使用不等式 \hbox{rank}(A-B)\ge\hbox{rank}A-\hbox{rank}B,可得

\hbox{rank}(A+B)=\hbox{rank}(A-(-B))\ge \hbox{rank}A-\hbox{rank}(-B)=\hbox{rank}A-\hbox{rank}B=r-k

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