每週問題 May 16, 2016

證明兩個矩陣的秩差的不等式。

Let A and B be m\times n matrices. Show that

|\hbox{rank}A-\hbox{rank}B|\le \hbox{rank}(A-B).

 
參考解答:

使用不等式 \hbox{rank}(X+Y)\le\hbox{rank}X+\hbox{rank}Y,可得

\hbox{rank}A=\hbox{rank}(A-B+B)\le \hbox{rank}(A-B)+\hbox{rank}B

因此,\hbox{rank}A-\hbox{rank}B\le\hbox{rank}(A-B)。根據對稱性,

\hbox{rank}B=\hbox{rank}(B-A+A)\le \hbox{rank}(B-A)+\hbox{rank}A=\hbox{rank}(A-B)+\hbox{rank}A

因此,\hbox{rank}B-\hbox{rank}A\le\hbox{rank}(A-B)。合併以上結果即得證。

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