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 are orthogonal matrices and is diagonal.