## Eigenvectors

Given an eigenvalue , nontrivial solutions of the homogeneous equation are called eigenvectors corresponding to . The null space of  is called the eigenspace corresponding to .

Let be scalars. Then the square matrix

is called the diagonal matrix with diagonal entries  .

In Matlab/Octave, the function diag(a) with column vector a returns the diagonal matrix  with diagonal entries given by the vector a. The command

[V,D] = eig(A)

returns a diagonal matrix  with eigenvalues on diagonal entries, and a matrix  whose column vectors are the corresponding eigenvectors.

EXAMPLE 4. Find a basis for the eigenspace corresponding to the value in each of the following.

1. with .
2. with .
3. with .

EXAMPLE 5. Find the eigenvalues and the corresponding eigenvectors for each of the following matrices.

Department of Mathematics