When we calculate singular values in Singular value decomposition we use the common eigenvalues (positive square roots) of $A^TA$ or $AA^T$, where $A$ is an $m\times n$ real matrix. We know that singular values of A may be repeated. Now I am trying to understand in which situation we will get repeated singular values of $A$.
I am thinking like that as we know $A^TA$ and $AA^T$ are both symmetric matrix, the singular values of $A$ will repeat when eigenvalues of $A^TA$ and $AA^T$ will repeat, i.e. symmetric matrix has repeated eigenvalues. But I did not find out when. Can any one help me to tell when eigenvalues of a symmetric matrix will repeat and also some example?
Thanks a lot.