每週問題 July 27, 2015

證明不存在恆定相似變換矩陣使任一矩陣相似於其轉置。

Prove that there is no nonsingular matrix P such that PAP^{-1}=A^T for every n\times n matrix A, n\ge 2.

 
參考解答:

使用反證法。假設存在一個可逆矩陣 P 使得每一個 n\times n 階矩陣 A 滿足 PAP^{-1}=A^T,也就是說有一個恆定相似變換矩陣 P 使得 A 相似於 A^T。對於任意 AB,使用三次上述性質,可得

\displaystyle  AB=P^{-1}A^TPP^{-1}B^TP=P^{-1}A^TB^TP=P^{-1}(BA)^TP=BA.

n\ge 2AB=BA 不總是成立,這樣便得到一個矛盾,證畢。

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