Let $M$ be a symmetric square matrix with complex coefficients, such that its imaginary part $N$ is positive definite.
Is it true that
$$A := {^t M} N^{-1} \overline M $$
has real coefficients?
(Here $^t M=M$ by symmetry).
I tested a random example, it seems to work. But writing the $ij$-coefficient of $A$ gives
$$\sum_{k,l} (^t M)_{il} (N^{-1})_{lk} \overline M_{kj}$$
so I'm not sure what to do.
Best Answer
Write $M=S+iN$, where $S$ is the (symmetric) real part of $M$. Then $$ A=M^TN^{-1}\overline{M}=MN^{-1}\overline{M}=(S+iN)N^{-1}(S-iN)=SN^{-1}S+N, $$ which is real.