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

To go a rather different route than the other answer here, you can use Cauchy-Binet to write $\det AA^T$ as a sum of squares. Cauchy-Binet tells us that $$ \det AA^T = \sum_{s\in S} \det A_{m,s} \det A^T_{s,m} = \sum_{s\in S} (\det A_{m,s})^2 $$

Where $S$ is the set of all size $m$ subsets of $[n]$, and $A_{m,s}$ is the submatrix from selecting only those $m$ rows from $A$ specified by $s$.

Of course to use this, you'd have to have proved Cauchy-Binet, which isn't too hard.

Answer 2

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 $$\det(AA') = \det(S^{-1}JS) = \det(S^{-1})\det(J)\det(S)=\det(J) \geq 0$$