Background
Assume we have a 2 columned matrix ${\bf P}$ and this matrix can be written as $${\bf P}= [ {\bf p_1 \,\,\,\, p_2}]$$ where ${\bf p_1}$ is the first column (vector) of ${\bf P}$ and ${\bf p_2}$ is the last column (vector) and assume we have another matrix ${\bf A}$.
I have seen a paper that expanded the following matrix multiplication ${\bf AP P^H A^H}$ (${\bf H}$ is hermitian: conjugate transpose) in the following way
$${\bf AP P^H A^H} = {\bf A}[ {\bf p_1 \,\,\,\, p_2}][ {\bf p_1 \,\,\,\, p_2}]^H {\bf A^H}={\bf Ap_1p_1^HA^H+ A{\bf p_2}{\bf p_2^H}A^H} $$
My problem
I am interested in a different matrix multiplication. Assume we have an additional matrix ${\bf W}$. I would like to expand matrix multiplication that is of the following form
$${\bf M}= {\bf W^HAP P^H A^HW}$$
so I thought of doing the same trick that is I defined $${\bf W}= [ {\bf w_1\,\,\,\, w_2}]$$
$${\bf M}={\bf W^HAP P^H A^HW}=[ {\bf w_1 \,\,\,\, w_2]^H} ({\bf Ap_1p_1^HA^H+ A{\bf p_2}{\bf p_2^H}A^H})[ {\bf w_1 \,\,\,\, w_2]} $$
$$=\begin{bmatrix} {\bf w_1^H}({\bf Ap_1p_1^HA^H+ A{\bf p_2}{\bf p_2^H}A^H}){\bf w_1} & {\bf w_1^H}({\bf Ap_1p_1^HA^H+ A{\bf p_2}{\bf p_2^H}A^H}){\bf w_2} \\ {\bf w_2^H}({\bf Ap_1p_1^HA^H+ A{\bf p_2}{\bf p_2^H}A^H}){\bf w_1} & {\bf w_2^H}({\bf Ap_1p_1^HA^H+ A{\bf p_2}{\bf p_2^H}A^H}){\bf w_2} \end{bmatrix}$$
My first question, is my derivation correct?
Assuming my derivation is correct, my second question I would like to solve for the following optimization matrix
$$\max_{{\bf w_1,w_2,p_1,p_2}} \log (|{\bf I}+ {\bf M}| )$$
it seems to me this is impossible to solve for, if yes can anyone help me with some ideas on how to simplify this complex problem, or give me hints ?
I assume the solution has to do with SVD decompostiion of $A$but I am not sure. Thanks