Number of Singular Values

Question

Is there any equation which describes or estimates the number of singular values of a Matrix $X$ ?
I found out that the number is equal to the number of eigenvalues of the Matrix $X^{*} X$, which are calculates as: $det( \lambda *I- X^{*} X)=0$. From these Eigenvalues I have to take the square roots and will get my singular values. But how do I see how many of those are equal to zero ?

Answer 1

That would be the rank of $X^*X$: the diagonalization of $X^*X$ is $$X^*X=P^{-1}DP$$ where $P$ is invertible, $D$ is diagonal with the eigenvalues on the diagonal.