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.
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.