My book lists out a bunch of steps for orthogonal diagonalization of a symmetric matrix, and one of them is
For each eigenvalue of multiplicity $>1$, find a set of $k$ linearly independent eigenvectors. If this set is not orthonormal, then apply the Gram-Schmidt orthonormalization process.
Let me start by saying that I don't actually know what the Gram-Schmidt orthonormalization process is, and I don't feel I should have to know it in order to understand my assigned reading since it was never actually taught in my course, but anyway I have a question:
Is this saying that a set of linearly independent eigenvectors corresponding to a certain eigenvalue may not be orthogonal, but that it's possible to make them that way? How could that be possible? The eigenvectors are always going to be at the same angle from each other no matter what scalar you multiply them by, right?