Re: [線代] eigenvector
※ 引述《mqazz1 (無法顯示)》之銘言：
: Every n*n matrix has n eigenvalues and n eigenvectors
: False
: ========
: (d) Each eigenvector of A is also an eigenvector of A^2
: True
2 2
Av ＝ λv => A v ＝ A(Av) ＝ A(λv) ＝ λ(Av) ＝ λ v.
v: eigenvector.
: (e) Each eienvector of an invertible matrix A is also
: an eigenvector of A^(-1)
: True
Note that if A is invertible, then 0 is not an eigenvalue of A.
-1 -1 -1
Av ＝ λv => v ＝ A (Av) ＝ A (λv) ＝ λ(A v)
-1 -1
=> A v ＝ λ v.
v : eigenvector, λ: eigenvalue.
: (f) If v is an eigenvector of an invertible matrix A,
: then cv is an eigenvector of A^(-1) for all nonzero scalars c
: True
-1 -1 -1 -1 -1 -1
Av ＝ λv => A v ＝ λ v => A (cv) ＝ c(A v) ＝ c(λ v) ＝ λ (cv).
v : eigenvector, λ: eigenvalue.
: 請問這些是為什麼?
: 謝謝
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