Re: [線代] eigenvalue一題
※ 引述《ntust661 (Enstchuldigung~)》之銘言：
: ※ 引述《xx52002 (冰清影)》之銘言：
: : Suppose A is a 3*3 real matrix such that A^3 = A^2 - A + I
: : (a) Find all possible eigenvalues of A.
: : (b) Determine the minimal and characteristic polynomial of A.
: : (c) Is A diagonalizable? Explain your answer.
: : 煩請各位解答 QQ
: 3 2
: λ - λ + λ - 1 = 0 (minimal and characteristic polynomial of A)
: λ = 1
: λ = i
: λ = -i
: 可以的^^ 因為特徵值對應的特徵向量互相線性獨立
min(λ)| (λ^3–λ^2 +λ–1)
∵A is a 3*3 real matrix
∴min(λ)= λ^3–λ^2 +λ–1 or λ–1
(1) min(λ)= λ^3–λ^2 +λ–1 ==> char(λ)=λ^3–λ^2 +λ–1
A is not diagonalizable over R
A is diagonalizable over C,
(2) min(λ)=λ–1 ==> char(λ)= (λ-1)^3
A is diagonalizable over R ( A ~ I_3 )
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