Matrix equation of the form $C A C^\intercal = D$

Question

Consider the following square matrix \begin{align} A = \left(\matrix{d & 0 & -\frac12 & 0 & 0 & 0 & 0 & 0 \\ 0 & d & -d+1 & -\frac12 & 0 & 0 & 0 & 0 \\ -\frac12 & -d+1 & d & 0 & -\frac12 & 0 & 0 & 0 \\ 0 & -\frac12 & 0 & d & -d+1 & -\frac12 & 0 & 0 \\ 0 & 0 & -\frac12 & -d+1 & d & 0 & -\frac12 & 0 \\ 0 & 0 & 0 & -\frac12 & 0 & d & -d+1 & -\frac12 \\ 0 & 0 & 0 & 0 & -\frac12 & -d+1 & d & 0 \\ 0 & 0 & 0 & 0 & 0 & -\frac12 & 0 & d }\right), \end{align} and the following rectangular matrix \begin{align} B = \left(\matrix{1 & \frac12 & \frac12 & 0 & 0 & 0 & 0 & 0 \\ 0 & \frac12 & \frac12 & 1 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 1 & \frac12 & \frac12 & 0 \\ 0 & 0 & 0 & 0 & 0 & \frac12 & \frac12 & 1 }\right), \end{align} then \begin{align} B A B^\intercal = \left(\matrix{d & 0 & -\frac14 & 0 \\ 0 & d & -d+\frac12 & -\frac14 \\ -\frac14 & -d+\frac12 & d & 0 \\ 0 & -\frac14 & 0 & d}\right). \end{align}

I am looking for a matrix $C$, the same size as $B$ such that \begin{align} C A C^\intercal = \left(\matrix{\frac{d}2 & 0 & -\frac14 & 0 \\ 0 & \frac{d}2 & -\frac{d}2+\frac12 & -\frac14 \\ -\frac14 & -\frac{d}2+\frac12 & \frac{d}2 & 0 \\ 0 & -\frac14 & 0 & \frac{d}2}\right). \end{align}

I get into a very complex system of equations in order to find $C$ that I am having a very hard time solving.

Does somebody see a way to profit from the sparsity and symmetries of $A$ to obtain $C$?

Answer 1

A suitable $C$ turns out to be pretty easy to find, probably much more so than was $B$. Observe that your target matrix is exactly $1/2$ times the upper-left $4\times4$ submatrix of $A$. So, we just need $C$ to pick out this submatrix and divide by $\sqrt2$. Now, left-multiplying by the $i$th row of the identity picks out the $i$th row of a matrix and right-multiplying by the $j$th column of the identity picks out the $j$th column. Fortunately, we want the same ranges of rows and columns, so the two selection matrices are each other’s transposes, which means we can take $$C = \frac1{\sqrt2}\begin{bmatrix}1&0&0&0&0&0&0&0\\0&1&0&0&0&0&0&0\\0&0&1&0&0&0&0&0\\0&0&0&1&0&0&0&0\end{bmatrix}.$$