# On a complex matrix $A$ such that $A A^T$ is real

Let $$A$$ be a complex invertible $$n\times n$$ matrix such that $$A A^T$$ is a real matrix. Does that imply that $$A^T A$$ is a real matrix too ?

Hint: Try (eg.) the matrix $$A = \left[ \begin{array}{cc} i & 0 \\ i & 1 \end{array} \right]$$.
• +1 Nice example. This matrix is in the form of $RD$ where $R$ is real and $D$ is a diagonal matrix such that $d_{kk}\in\{1,i\}$ for each $k$. Apr 11 at 5:04