[Math] is determinant of A times A transposed bigger than or equal to zero

Question

We have an m by n matrix A of real numbers where n is bigger than m.
Prove that determinant of A times A transposed is bigger than or equal 0.

Answer 1

Since $AA'$ is positive semidefinite, in the eigendecomposition (or Jordan Canonical form) of $AA' = S^{-1}JS$, the diagonal matrix $J$ only has positive values on the diagonal, hence $$\det(AA') = \det(S^{-1}JS) = \det(S^{-1})\det(J)\det(S)=\det(J) \geq 0$$