I have been given a $3×3$ Hermitian matrix $H$. I am asked to come up with a non singular matrix $P$ such that $D=P^{T}H\overline{P}$ where $P^{T}$ represents transpose of matrix $P$ while $\overline{P}$represents conjugate of $P$.
As $H$ is a Hermitian matrix hence it can be diagonalized by a unitary matrix $P$. This means that we can write $H$ as follows:
$H=P^{-1}DP$ or $D=PHP^{-1}$
As $P$ is unitary, we have: $P^{\theta}=P^{-1}$ where $P^{\theta}$ represents conjugate transpose of $P$. Thus we have:
$D=PHP^{\theta}$
But question wants me to have $P^{T}$ in place of $P$ and $\overline{P}$ in place of $P^{\theta}$. I do not know how to proceed from this point. Please help me.
Notice that $\overline{H}$ is also Hermitian. Diagonalizing $\overline{H}$, using the notation in the OP, yields $$D = Q\overline{H}Q^{\theta}$$ for some unitary $Q$. Now define $P := Q^{\theta}$ and take complex conjugates to obtain $$D = P^TH\overline{P},$$ noting that $D = \overline{D}$ as all eigenvalues of Hermitian matrices are real.