What will the rank be, if a matrix multiply its transpose?

Question

Let's say we got a matrix $A_{n\times k}$ and $n>k$, and the column rank of $A$ is $k-1$ if we pre- and postmultiply its transpose, i.e. $A^TA$ and $AA^T$.

Is the rank of $A^TA$ and $AA^T$ still $k-1$?

Answer 1

For a real matrix $A$, $\|A x\|^2 = x^T A^T A x$ so $A x = 0$ if and only if $A^T A x = 0$. Thus the ranks of $A^T A$ and $A$ are always equal. Similarly, $\text{rank}(A) = \text{rank}(A^T) = \text{rank}(A A^T)$.