# Find a Matrix P that diagonalizes A

1. Apr 4, 2014

### mpittma1

1. The problem statement, all variables and given/known data
Find a Matrix P that diagonalizes A

2. Relevant equations

A = $$\begin{pmatrix} 2 & 0 & -2\\ 0 & 3 & 0\\ 0 & 0 & 3 \end{pmatrix}$$

3. The attempt at a solution

Well right off the bat we know that this is an upper triangular matrix so the eigenvalues are the entries along the main diagonal of A.

So λ = 2, 3, 3

But if an n x n matrix A has n distinct eigenvalues, then A is diagonalizable.

In this case we only have 2 distinct eigenvalue so it shouldnt be diagonalizable.....

P = $$\begin{pmatrix} -2 & 0 & 1\\ 0 & 1 & 0\\ 1 & 0 & 0 \end{pmatrix}$$

What is the proper way to start the problem to find this matrix P??????

Thank You for any help in advance.

2. Apr 5, 2014

### Staff: Mentor

If you check, you'll see that the eigenspace of λ = 3 is of dimension 2, so there are two eigenvectors for this eigenvalue.

When a matrix has repeated eigenvalues, there is some terminology that distinguishes between algebraic multiplicity vs. geometric multiplicity. I think this matrix is a case of geometric multiplicity.

To find your matrix P, the important thing is to get three linearly independent eigenvectors, not whether there are three distinct eigenvalues.