In a lot of texts I have seen involving the singulare value decomposition, it only says, that there are as many nonzero singular value as is the rank of the matrix $A$, which is to be decomposed.
Now I have looked at different examples throughout the net and everywhere these singular values are distinct. But as I understood the proof of the singular value decomposition, there are indeed $r$ singular values (if $rank(A)=r)$, but they may appear more than once: To be specific, they appear as often as the algebraic multiplicity of the eigenvalue, which corresponds to this singular value (since the sum of all algebraic multiplicities of all nonzero eigenvalues is $r$).
Is this correct ? Could anyone give me an example of a matrix $A$, such that this happens, i.e. such that $A^*A$ has eigenvalues with algebraic multiplicity $>1$ ?