[線代] Friedberg證矩陣版Cayley-Hamilton定理

[cyt147 (大叔)]

大家早安!有個證明想請教各位，謝謝。


下面是C.-H.定理:

Let T be a linear operator on a finite-dimensional vector space V over F and

let f(t) be the characteristic polynomial of T. Then f(T)=T_0, the zero

operator.


Now I want to prove its corollary for matrices. That is, I want to prove the

following theorem by using the C.-H. Theorem:


Let A be an nxn matrix and let f(t) be the characteristic polynomial of A.

Then f(A)=O, the zero matrix.

proof: Since the characteristic polynomial of L_A(the left-multiplication

       transformation for A) is f(t), by the C.-H. Theorem, f(L_A)=T_0.

       ..............To be continued.


雖然不太確定，但 L_{f(A)}=f(L_A)應該是正確的，接下來的問題就是證:

若y為任意nx1矩陣，且f(A)y=O_{nx1}，則f(A)=O

請問各位，我該如何解決這個問題呢?感謝!

===================提問結束，補證明如下============================


首先，請複習Friedberg第2章，這邊有提到左乘變換的一些性質，用這些性質

，我們可以輕易證得L_{f(A)}=f(L_A)，既然f(L_A)為零變換，我們可以導出

L_{f(A)}=L_{O}，接下來請看Theorem 2.15.(b)，結論:f(A)=O
※ 編輯: cyt147 (140.122.140.36), 07/12/2017 12:53:54
→ LiamIssac   : independence？
本提問結束了。你好，請問哪邊有問題嗎?
※ 編輯: cyt147 (140.122.140.36), 07/12/2017 13:38:40