Eigenvalue and eigenvectors

An eigenvector of an

n \times n

matrix

A

is a non-zero vector

x

such that

Ax= \lambda x

, for some scalar

\lambda

. The scalar

\lambda

is called the eigenvalue.

We say the eigenvector

x

corresponds to the eigenvalue

\lambda

.

Given an eigenvalue

\lambda

of matrix

A

, we can find a corresponding eigenvector

x

by solving

(A-\lambda I)x=0

And finding a non-trivial solution

x

.

The eigenspace is the null space of the matrix

A-\lambda I

. In other words, the eigenspace is a set of all solutions for the equation

(A-\lambda I)x=0

Of course, we can find the basis for the eigenspace by finding the basis of the null space of

A-\lambda I

1.
Verifying Eigenvectors
Let . Is an eigenvector of $A$ ? If so, find the eigenvalue. What about ?

3.
Verifying Eigenvalues and finding a corresponding eigenvector
Let . Is $\lambda=1$ an eigenvalue of $A$ ? If so, find a corresponding eigenvector.

4.
Finding a Basis for the Eigenspace
Find a basis for the corresponding eigenspace if:

5.
Proof Related to Eigenvalues and Eigenvectors
Prove that if $A^2$ is the zero matrix, then the only eigenvalue of $A$ is 0.

6.
Let $\lambda$ be an eigenvalue of an invertible matrix $A$ . Then $\lambda ^{-1}$ is an eigenvalue of $A^{-1}$ .