I understand that for each of the $N$ eigenvalues (regardless of repeated or not) of an $N\times N$ symmetric matrix, the algebraic multiplicity and geometric multiplicity are equal. This means if an eigenvalue is repeated $M$ times, then we have exactly $M$ orthogonal eigenvectors associated with this eigenvalue.
What I haven't figured out are:
- Each entry of my $400\times400$ symmetric matrix is from $0$ to $1$ (both inclusive). I notice that many eigenvalues crowd at $0$, $-1$ and $1$, if they are not actually equal. On the other hand, eigenvalues elsewhere are well dispersed (figure below). Does this tell me anything about my symmetric matrix?
- In fact, entries of my matrix distribute weirdly between $0$ and $1$ in the sense that the predominant majority of the values (after I exlude zero entries) distribute between $0$ and $0.1$ (figure below). Maybe this has something to do with the result above?
- If this is not a numerical issue (i.e., the eigenvalues are really repeated), when does a symmetric matrix have repeated eigenvalues? Or what can we say about such a matrix?