Matrices may be used to permute the order of elements in a set. For example:
$ \begin{bmatrix} 0 & 0 & 0 & 1 \\ 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \end{bmatrix} \times \begin{bmatrix} x \\ y \\ z \\ w \end{bmatrix} = \begin{bmatrix} w \\ x \\ y \\ z \end{bmatrix} $
My problem is that I need to permute values in different matrices:
$ A = \begin{bmatrix} 0 & 0 & x & 0 & 0 \\ x & x & x & x & x \\ x & 0 & 0 & 0 & x \\ 0 & 0 & 0 & 0 & 0 \end{bmatrix} $
$ B = \begin{bmatrix} 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & y & 0 & 0 \\ y & y & y & y & y \\ y & 0 & 0 & 0 & y \end{bmatrix} $
$ C = \begin{bmatrix} z & 0 & 0 & 0 & z \\ 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & z & 0 & 0 \\ z & z & z & z & z \end{bmatrix} $
$ D = \begin{bmatrix} w & w & w & w & w \\ w & 0 & 0 & 0 & w \\ 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & w & 0 & 0 \end{bmatrix} $
Thus given the permutation matrix (from above) and the following pattern matrix how would you change all the $x$ in $A$ to $w$, all the $y$ in $B$ to $x$, all the $z$ in $C$ to $y$, and all the $w$ in $D$ to $z$:
$ \begin{bmatrix} 0 & 0 & 1 & 0 & 0 \\ 1 & 1 & 1 & 1 & 1 \\ 1 & 0 & 0 & 0 & 1 \\ 0 & 0 & 0 & 0 & 0 \end{bmatrix} $