Minimum value trace of A*Atranspose For any matrix $A$, let $A^T$ denote its transpose matrix.
What is the minimum value of $\mathrm{trace}(AA^T)$ for an $n \times n$ non-singular matrix $A$ with integer entries?

When $A$ is nonsingular, $AA^T$ is positive definite. Hence it has a positive diagonal. But $AA^T$ is also an integer matrix. Hence its diagonal entries are positive integers, meaning that $\operatorname{tr}(AA^T)\ge n$. Obviously, tie occurs when $A=I$.
