There is a rule that says:
for an m x m matrix.
are singular values. It doesn't specify whether they are just singular values or non-zero singular values.
Does this mean just the non-zero singular values?
Because that means a determinant I'm working on is zero, not 12.
can someone please tell me?
I just thought about it some more though.
If for example, a 2 x 2 matrix had only one non-zero singular value it means that it's rank one, not full rank, which mean's it's not invertible, which means that it has a 0 determinant.
So I'm thinking that you have to go with how the equation reads and include all singular values non-zero or not.
Do you agree?
"Singular values" are just the eigenvalues of the matrix. Any matrix is similar to a matrix in either diagonal form or Jordan normal form where the determinant is just the product of the values on the diagonal. And the values on the diagonal are the eigenvalues.
And if a matrix has any 0 eigenvalue, then its determinant is 0.
(If the determinant were the product of the "non-zero" singular values, then NO matrix would have 0 determinant!)