2

Call a matrix $A$ weakly square iff $\mathrm{det}(A^\top A) = \mathrm{det}(A A^\top)$. Then clearly,

  • every square matrix is weakly square, and
  • every zero matrix is weakly square.

Question. Are these the only examples of weakly-square matrices?

Remark. I got the idea from Donald Reynolds answer here.

Community
  • 1
goblin GONE
  • 67,744

1 Answers1

2

A non-square matrix $A$ is weakly square if and only if neither $A$ nor $A^T$ has full rank, which is to say iff $\operatorname{rank}(A)<\min\{m,n\}$.

The key to this observation is to note that $$ \operatorname{rank}(A^TA)= \operatorname{rank}(A)= \operatorname{rank}(A^T)= \operatorname{rank}(AA^T) $$

Ben Grossmann
  • 225,327
  • Could you elaborate a bit? I don't really get it: in particular, why does that chain of equalities hold, and why does it imply the statement under question? – goblin GONE Jan 01 '16 at 14:00
  • A square matrix has non-zero determinant iff it's rank is less than its size. The equalities are sufficiently well known that you can look them up easily. For the first, one shows that $A^TAx=0\iff Ax=0$ by considering $x^TA^TAx$. – Ben Grossmann Jan 01 '16 at 14:06