# Math Help - Lin Alg Proof

1. ## Lin Alg Proof

Given that L is an eigenvalue of an invertible matrix A, prove that 1/L is an eigenvalue for A^(-1). (NOTE: need proof for A in general; giving one example will not suffice)

2. Originally Posted by caeder012
Given that L is an eigenvalue of an invertible matrix A, prove that 1/L is an eigenvalue for A^(-1). (NOTE: need proof for A in general; giving one example will not suffice)
L is an eigenvalue of A, so there exists an non-zero vector X such that:

AX=LX

but A is invertible, so A^{-1} exists, and so:

A^{-1}[AX]=A^{-1}LX=L A^{-1}X,

so:

X= L A^{-1}X,

or (as L cannot be 0):

A^{-1}X=(1/L)X.

Hence (1/L) is an eigenvalue of A^{-1}

RonL