[代數] eigenvector
做考古題做到的ㄧ題，做幾行就寫完了感覺不太符合配分
所以想上來請教一下
1.Let S and T be linear transformations on a finit-dimensional vector
space over C.Suppose ST=TS.Show that they have a common eigenvector.
2.(p-1)!≡p-1 mod(1+2+...+p-1) if p is a prime number.
第一題我的解法是:假設x=\=0 是S的ㄧ個eigenvector，而λ是相對應
的eigenvalue
所以ST(x)=TS(x)=T(λx)=λT(x)
＝〉T(x)也是一個相對應λ的eigenvector,所以T(x)=ax,where x屬於R
所以x就是S,T共同的eigenvector.
不知道是不是哪裡有盲點，這樣解的話，對所有的eigenvector都對吧
所以想請幫忙看一下是不是哪裡解錯了
而第二題不知道怎麼解，所以想請教一下
感謝
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