[理工][線代]對角化問題
看了黃子嘉老師的線性代數下冊
對角化有個定理不是很懂，請神人幫忙解惑，謝謝
Q1:thm.5-21 p.5-64
假設 A : n x n，pA(x)在F中可分解且pA(x)=(λ1-x)(λ2-x)...(λn-x)，則
(1)det(A)=λ1λ2λ3...λn
(2)tr(A)=λ1+λ2+λ3+...λn
想問說Ａ可分解是假設的還是所有nxn matrix都可以分解?
因為我看下面的練習題都是直接當成矩陣可分解.
所有nxn矩陣的det和tr都可以這樣算?
Q2:另外，旁邊p5-65的台大考題:
If λ1,λ2,λ3,λ4,λ5 are all the eigenvalues of the matrix
／ ＼
︱ 4 -1 0 0 0 ︱
︱ ︱
A =︱ -1 3 -1 0 0 ︱
︱ ︱
︱ 0 -1 3 -1 0 ︱
︱ ︱
︱ 0 0 -1 3 -1 ︱
︱ ︱
︱ 0 0 0 -1 3 ︱
︱ ︱
＼ ／
Find (λ1)^2+(λ2)^2+(λ3)^2+(λ4)^2+(λ5)^2.
我不懂為什老師書上寫說:
"因為A具eigenvalue λ1,λ2,λ3,λ4,λ5
所以A^2具eigenvalue (λ1)^2...(λ5)^2"
Q3: 當有特徵方程式有重根時，他的eigenvalue要寫的和重根數一樣多還是只要寫一個?
eg: pA(x) = (x-1)^2,則所有的的eigenvalue為1,還是1,1?
老師書上兩種都有寫過
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