# Find orthogonal P and diagonal matrix D

1. Jun 6, 2010

### phamdat1202

1. The problem statement, all variables and given/known data

A= [1 -1 0]
[-1 2 -1]
[0 -1 1]
find orthogonal matrix P and diagonal matrix D such that P' A P = D

2. Relevant equations

3. The attempt at a solution
i got eigenvalues are 0, 1, 3 which make D=[0 0 0; 0 1 0; 0 0 3]
how to find P. because in my solution they mentioned about normalised eigenvectors.

2. Jun 6, 2010

### vela

Staff Emeritus
Find the eigenvectors. The columns of P are the eigenvectors.

3. Jun 6, 2010

### phamdat1202

i know after i got eigenvalues, i can find eigenvectors which is P.
my question is that in my solution, P are normalised eigenvectors. why they use normalised eigenvector instead of the eigenvector?

4. Jun 6, 2010

### D H

Staff Emeritus
If you used unnormalized eigenvectors the diagonalization equation is P-1AP=D instead of P*AP=D. The inverse is particularly easy to find with normalized eigenvectors. If P is constructed from normalized, orthogonal eigenvectors then P will be an orthogonal matrix, making P-1=P*.

5. Jun 6, 2010

### phamdat1202

thanks, got it