Let
$$B := \begin{bmatrix} j H & kH \\ kH & H\end{bmatrix}$$
where $H$ is a circulant matrix and it is symmetric and non-invertible, and $j, k$ are scalars. Let
$$A := (B+C)D $$
where $C$ is a skew-symmetric matrix and $D$ is a positive definite matrix. Everything is real-valued.
I would like to find the trace of $D$ in terms of $A$, $H$, $j$, $k$. Could anyone please help me to solve the question?
several observations:
Firstly: B is symmetric and non-invertible, and is a Kronecker product of $\begin{bmatrix} j & k \\ k & 1\end{bmatrix}$ and $H$
Secondly: $\operatorname{Tr}(A)=\operatorname{Tr}(BD)$