Find orthogonal P and diagonal matrix D

[phamdat1202]

Homework Statement

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

Homework Equations

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.

 

[vela]

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

 

[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?

 

[D H]

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^*.

 

[phamdat1202]

thanks, got it