# Let A be an invertible nxn matrix and v an eigenvector of A with associated eigenvalue λ.?

a) Is v an eigenvector of A^3? if so, what is the eigenvalue?

b) Is v an eigenvector of A^-1? if so, what is the eigenvalue?

c) Is v an eigenvector of A + 2I? if so, what is the eigenvalue?

d) is v an eigenvector of 7A? if so, what is the eigenvalue?

e) Let A be an nxn matrix and let B=A-αI for some scalar α. How do the eigenvalues of A and B compare?

Relevance

for v to be an eigenvector with eigenvalue λ

A v = λ v

a) A v = λ v

multiply by A

A^2 v = A λ v = λ A v = λ λ v = λ^2 v

A^3 v = A λ^2 v =λ^2 A v = λ^3 v

so v is an eigenvector of A^3 and λ^3 is the associated eigenvalue

b) A v = λ v

left multiply by A^-2

A^-2 A v = A^-2 λ v

A^-1 v = λ A^-2 v = (λ A^-2) v

for v to be an eigenvector of A^-1 then A^-2 must = I the unit matrix

so v is not in general an eigenvector of A^-1

c) A v = λ v

add 2 I v to both sides

Av + 2 I v = λ v + 2 I v or noting that 2 I v = 2 v on the right hand side

(A+2 I ) v = (λ + 2 ) v

so v is an eigenvector of Av + 2 I v with associated eigenvalue of λ + 2

d) A v = λ v

7 A v = 7 λ v

(7A) v = (7 λ) v

so v is an eigenvector of 7A and has eigenvalue of 7 λ

e) A v = λ v and

B=A-αI

B v = A v - α v

B v = λ v - α v = (λ - α) v

so the eigenvalue of B called λb is

λb = λ - α