The command MatrixMultiply(A, B, rc, R) returns the product of A and B mod the saturated ideal of rc.

The result is viewed as a matrix with coefficients in the total ring of fractions of R/I where I is the saturated ideal of rc.

The implementation is based on the method proposed in the paper "On {W}inograd's Algorithm for Inner Products" by A. Waksman.

It is assumed that rc is strongly normalized.

This command is part of the RegularChains[MatrixTools] package, so it can be used in the form MatrixMultiply(..) only after executing the command with(RegularChains[MatrixTools]).  However, it can always be accessed through the long form of the command by using

$\mathrm{with}\left(\mathrm{RegularChains}\right)\:$$\mathrm{with}\left(\mathrm{ChainTools}\right)\:$$\mathrm{with}\left(\mathrm{MatrixTools}\right)\:$

$R≔\mathrm{PolynomialRing}\left(\left[y\,z\right]\right)$

$R≔\mathrm{polynomial\_ring}$

$\mathrm{rc}≔\mathrm{Empty}\left(R\right)$

$\mathrm{rc}≔\mathrm{regular\_chain}$

$\mathrm{rc}≔\mathrm{Chain}\left(\left[z^4+1\,y^2-z^2\right]\,\mathrm{rc}\,R\right)\:$

$\mathrm{Equations}\left(\mathrm{rc}\,R\right)$

$\left[y^2-z^2\,z^4+1\right]$

$m≔\mathrm{Matrix}\left(\left[\left[1\,y+z\right]\,\left[0\,y-z\right]\right]\right)$

$m≔\left[\begin{array}{cc}1& y+z\\ 0& y-z\end{array}\right]$

$\mathrm{mim}≔\mathrm{MatrixInverse}\left(m\,\mathrm{rc}\,R\right)$

$\mathrm{mim}≔\left[\left[\left[\left[\begin{array}{cc}1& 0\\ 0& \frac{z^3}{2}\end{array}\right]\,\mathrm{regular\_chain}\right]\right]\,\left[\left[''noInv''\,\left[\begin{array}{cc}1& y+z\\ 0& y-z\end{array}\right]\,\mathrm{regular\_chain}\right]\right]\right]$

$\mathrm{m1}≔\mathrm{mim}\left[1\right]\left[1\right]\left[1\right]$

$\mathrm{m1}≔\left[\begin{array}{cc}1& 0\\ 0& \frac{z^3}{2}\end{array}\right]$

$\mathrm{rc1}≔\mathrm{mim}\left[1\right]\left[1\right]\left[2\right]$

$\mathrm{rc1}≔\mathrm{regular\_chain}$

$\mathrm{MatrixMultiply}\left(\mathrm{m1}\,m\,\mathrm{rc1}\,R\right)$

$\left[\begin{array}{cc}1& 0\\ 0& 1\end{array}\right]$

A. Waksman "On Winograd's Algorithm for Inner Products." IEEE Transactions On Computers, C-19, (1970): 360-361.