[線代] similar跟Cayley-Hamilton

[mqazz1 (無法顯示)]

A invertible matrix can be similar to a singular matrix
        False

A symmetric matrix can be similar to a nonsymmetric matrix
        True

請問這兩個要怎麼證?

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A = [ 2  3 ]    求A^10          答案[ -6232 -3353 ]

這個 PA(x) = x^2 - 4x + 5

可是用長除法去找餘式有點麻煩

請問有更好的解法嗎?

謝謝

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◆ From: 192.192.13.101
推 jacky7987   :第一個,If true, suppose A is nonsingular, B is
→ jacky7987   :singular and exists a invertible matrix P s.t.
→ jacky7987   :A=P^{-1}BP  then det(A)=0
→ jacky7987   :similar 有不少東西都一樣 det tr jordan form等