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Cayley–Hamilton theorem是否可逆？

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Cayley–Hamilton theorem是否可逆？ 即是問以下的statement是否成立？ For a matrix polynomial equation p(A) = 0, the only matrices A can satisfy this matrix polynomial equation p(A) = 0 is that it exist some of its eigenvalue(s) λ satisfy this polynomial equation p(λ) = 0. 請詳細解釋。

最佳解答:

If p(A)=0, t is any eigenvalue of A, then 0=p(A)v=p(t) v, so p(t)=0 請問d大您問什麼問題? 2010-03-01 14:39:11 補充： Let λ be an eigenvalue of A with eigenvector v (Av=λv), then p(A)v=p(λ)v. While p(A)=0, so p(λ)v=0, p(λ)=0, ie. eigenvalue(s) of A must be root(s) of p(x)=0. For instance, if A satisfies matrix eq. X^2-4X+3I=0, then eigenvalue(s) of A must be 1 or 3.

其他解答:

應該是問 P( A ) = 0→P( λ )會不會等於0