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Eigenspace

$Eigenvalue, eigenvector$

λ

is an eigenvalue of

f

and

v

is an eigenvector of

f

:

f (v) = λ v

$Characteristic polynomial$

Characteristic polynomial of

A

:

e_{A} (λ) = ∣ A - λ E ∣

$Cayley-Hamilton theorem$

e_{A} (A) = 0

$Eigenspace$

Eigenspace of

λ

in

A

:

W_{λ} = {v ∣ A v = λ v}

$Corollary$

Eigenvectors are linear independent

$Projection$

V = W \oplus W^{⊥} x = x_{1} + x_{2}

Projection to

W

:

A : x \to x_{1}