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As the title says. Is the matrices $A^*A$ and $AA^*$ hermitian (symmetric if $A$ is real)?

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Hint: $(AB)^* = B^* A^*$ – Nate Eldredge Jan 16 '12 at 19:15
A matrix $M$ is Hermitian if $M^* = M$. Take $M = A^* A$. Does $(A^* A)^* = A^* A$? Do you know how to express $(AB)^*$ in terms of $A^*$ and $B^*$? – Michael Joyce Jan 16 '12 at 19:16
Thanks! These hints really helped. Remember this from the lectures now. How could I not see this? – Godisemo Jan 16 '12 at 19:24
up vote 2 down vote accepted

Check the definition of hermitian. This is not too hard, you just have to use that $(AB)^*=B^*A^*$. You don't need that $A$ is invertible in this proof, the statement is even true for $A$ not a square.

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