It is often useful to be able to transform a matrix into a matrix with non zero entries only on the diagonal. If the matrix A arises in a system of differential equations, the system often becomes easier to solve, and a phase space diagram becomes easier to sketch.

We start by finding the eigenvalues of the matrix. We the eigenvalues, we can find the eigenvectors. Then we construct the matrixwith columns consisting of the eigenvectors. The matrixwill be the matrix we seek.

Example: Diagonalize the matrix

First find the eigenvalues of the matrix

Now find the eigenvectors:

is an eigenvector.

Hence is an eigenvector.

Takethenand