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 ?
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Hint: Try (eg.) the matrix $A = \left[ \begin{array}{cc} i & 0 \\ i & 1 \end{array} \right]$.
answered Apr 11 at 0:25
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$\begingroup$ +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 $k$. $\endgroup$– user1551Apr 11 at 5:04