[2] It is a bound on the determinant of a matrix whose entries are complex numbers in terms of the lengths of its column vectors.
In geometrical terms, when restricted to real numbers, it bounds the volume in Euclidean space of n dimensions marked out by n vectors vi for 1 ≤ i ≤ n in terms of the lengths of these vectors ||vi ||.
Then So, the determinant of a positive definite matrix is less than or equal to the product of its diagonal entries.
[2][5] The result is trivial if the matrix N is singular, so assume the columns of N are linearly independent.
The general result now follows: To prove (1), consider P =M*M where M* is the conjugate transpose of M, and let the eigenvalues of P be λ1, λ2, … λn.