Eigenvalues and Eigenvectors

An eigenvectorof a matrixis such that if the vector is multiplied by the matrix, the result is a multiple of the vector. Eigenvectors are special and have many applications in may areas. We ,ay write

In this equationis a constant called the eigenvalue.

The procedure for finding eigenvectors and eigenvalues is quite simple. IfthenThis means that the matrixhas zero determinant. We can solveand solve this equation to find values ofIn general several values of %lambda may be found. Each value ofgives value to at least one eigenvector, and different eigenvectors give rise to different eigenvalues.

Example: Find the eigenvalues and eigenvectors for the matrix

We obtain

Solving this givesor

We find the eigenvectorsby solving

For

Henceand an eigenvector is

For

Henceand an eigenvector is

Notice that any vectors of the formsandare eigenvectors. We choose values ofto make the eigenvectors as simple as possible.