Eigenvalues and Eigenvectors

Question 1

eigenvector/ eigenvalue

Answer 1

let A be a square nxn matrix. A non-zero n vector v is a eigenvector of A with eigenvalue λ if Av = λv
or (A-Iλ)v = 0

Question 2

whens non-zero v

Answer 2

iff det(A - Iλ) = 0 which implies A - I λ non-invertible

Question 3

characteristic polynomial

Answer 3

Pa(t) = det(A - tI)
Pa(λ) = 0 to find λ (eigenvalues)

Question 4

algebraic multiplicity

Answer 4

P(t) = C(t-λ1)^k1(t-√2)^k2….(t-λp)^kp
∑ki = N , λi ≠ λj
λi has algebraic multiplicity ki

Question 5

eigenspace

Answer 5

for eigenvalue λ with algebraic multiplicity k the eigenspace V is the P dimensional vector space spanned by these eigenvectors
Vλ = ker (A - Iλ)
(when A - Iλ = 0)

Question 6

algepraic multiplicity k meaning

Answer 6

if λ has algebraic multiplicity k then there are P linearly independent eigenvectors where 1<= p<= k

Question 7

geometric multiplicity

Answer 7

P = dimV is the geometric multiplicity of λ