I'm given a matrix

A = -3 0 -4

0 5 0

-4 0 3

I have to orthogonally diagonalize the matrix and give the orthogonal matrix P and the diagonal matrix D. I have no idea how to do this.

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- Jul 23rd 2009, 09:53 AMlovestoskateHow do I orthogonally diagonalize a matrix?
**I'm given a matrix**

A = -3 0 -4

0 5 0

-4 0 3

I have to orthogonally diagonalize the matrix and give the orthogonal matrix P and the diagonal matrix D. I have no idea how to do this. - Jul 23rd 2009, 10:33 AMHallsofIvy
This is a "symmetric" matrix and so must have 3 independent eigenvectors. In fact, eigenvectors corresponding to different eigenvalues must be orthogonal.

So do this: find the eigenvalues and the corresponding eigenvectors. If there is a "double" or "triple" eigenvalue, you can find vectors in that eigenspace that are orthogonal. Divide each eigenvector by its length to get unit vectors.

Form matrix Q using those vectors as columns. Then Q and $\displaystyle Q^{-1}$ are orthogonal matrices and $\displaystyle Q^{-1}PQ$ is diagonal.