I know that when expressing a matrix $A$ in SVD, the determinant of $A$ can be calculated by finding the product of the singular values $\sigma_1 *... *\sigma_n $. My question is, does this hold true only for non-zero values of $\sigma_i$? So for instance, I am given the singular matrix given below:
$\Sigma$ = $$ \begin{bmatrix} 4 & 0 & 0 & 0 \\ 0 & 2 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 0 \\ \end{bmatrix} $$
Would the determinant in this case be 0? Or would we just multiply the non-zero singular values to get a determinant of $8$?