2
$\begingroup$

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 ?

$\endgroup$

1 Answer 1

4
$\begingroup$

Hint: Try (eg.) the matrix $A = \left[ \begin{array}{cc} i & 0 \\ i & 1 \end{array} \right]$.

$\endgroup$
1
  • $\begingroup$ +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 $k$. $\endgroup$
    – user1551
    Apr 11 at 5:04

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.