Example of a symmetric matrix which doesn’t have orthogonal eigenvectors

eigenvalues-eigenvectorslinear algebramatricessymmetric matrices

I'm looking for an example of a symmetric matrix $A$ which doesn't have orthogonal eigenvectors.

Here's what I tried: I was able to prove that the eigenvectors corresponding to each distinct eigenvalue of a symmetric matrix are orthogonal. So, I realise that the example I'm looking for is a symmetric matrix with at least one repeated eigenvalue for which there are no orthogonal eigenvectors. But I'm not sure how to construct such an example.

Best Answer

So, I realise that the example I'm looking for is a symmetric matrix with at least one repeated eigenvalue for which there are no orthogonal eigenvectors.

If you ever get two of more linearly independent eigenvectors corresponding to the same eigenvalue, you can apply the Gram Schmidt process and end up with orthogonal vectors... which will continue being eigenvectors!

Thus, it is not possible to get any symmetric matrix which doesn't have orthogonal eigenvectors. (Note that I'm not saying that all eigenvectors will be orthogonal but that you can always find orthogonal ones.)

Related Question