## Eigenvectors

## Definition. If A is an

nxnmatrix, then a nonzero vector x in R^{n}is called an eigenvector of A if Ax is a scalar multiple of x; in other words,

## for some scalar . The scalar is called an eigenvalue of A and x is the eigenvector corresponding to .

## Theorem: If A is an

nxnmatrix, then the following are equivalent.## a) is an eigenvalue of A.

## b) The system of equations (

I- A)x = 0 has nontrivial solutions.## c) There is a nonzero vector x in R

^{n}such that Ax = x.## d) is a real solution of the characteristic equation det(

I- A) = 0.

A.8a