Consider first the bivariate normal case in which the variables are
perfectly correlated. Then the density is zero everywhere except on
some line, where it is univariate normal.
The n-variate case is analogous. If the covariance matrix has rank k
then the density is zero everywhere except in some k-dimensional
hyperplane, where it is k-variate normal.
Let x be such an n-vector, with mean vector m and covariance matrix S.
Let P be the n by k matrix of unit-length eigenvectors corresponding
to the nonzero eigenvalues of S, and let Q be any n by (n-k) matrix
whose columns span the null space of S. Let y = P'(x-m), and let z =
Q'(x-m). If z'z > 0 then x is not in the hyperplane, and the density
is zero. If z = 0 then x is in the hyperplane, where its conditional
density is that of y, which is k-variate normal with mean vector 0
and covariance matrix P'SP (which is diagonal; the variances are the
nonzero eigenvalues of S).
It's the product of the non-zero singular values. Ray gave a good
geometric explanation of why this works. It's the determinant of the
full support basis vectors of the covariance matrix.
Go back to that thread and find the sentence
Now replace inv(C) with C# and det(C) with the product s1*s2*...sr.
Hope this helps.
Greg