While real symmetric matrices have real eigenvalues, their eigenvectors do not have to be real. If I am interpreting the conversation correctly, THIS math.stackexchange thread seems to suggest that it is, nonetheless, possible to choose a basis such that ALL of the eigenvectors are real. Is this true?
If so, how would I do this?
As a concrete example, I have a 448 x 448 real symmetric matrix. I compute the eigenvalues and eigenvectors using MATLAB and 24 of the eigenvectors that it returns are complex, all the rest are real. How can I make these 24 complex eigenvectors real?