These commands provide, within the DifferentialGeometry environment, the basic arithmetical operations for Matrices or Vectors of: vectors, differential forms, or tensors.  They are particularly useful for curvature calculations for connections on principle bundles of matrix groups.

These commands are part of the DifferentialGeometry:-Tools package, and so can be used in the form described above only after executing the commands with(DifferentialGeometry) and with(Tools) in that order.

$\mathrm{with}\left(\mathrm{DifferentialGeometry}\right)\:$$\mathrm{with}\left(\mathrm{Tools}\right)\:$

$\mathrm{DGsetup}\left(\left[x\,y\,z\right]\,M\right)\:$

$A≔\mathrm{Vector}\left(\mathrm{evalDG}\left(\left[\mathrm{dx}-\mathrm{dy}\,\mathrm{dy}\+\mathrm{dx}\right]\right)\right)$

$\left[\begin{array}{c}\mathrm{\_DG}\left(\left[\left[''form''\,M\,1\right]\,\left[\left[\left[1\right]\,1\right]\,\left[\left[2\right]\,-1\right]\right]\right]\right)\\ \mathrm{\_DG}\left(\left[\left[''form''\,M\,1\right]\,\left[\left[\left[1\right]\,1\right]\,\left[\left[2\right]\,1\right]\right]\right]\right)\end{array}\right]$

$B≔\mathrm{Vector}\left(\mathrm{evalDG}\left(\left[\mathrm{dx}\+2\mathrm{dy}\,\mathrm{dx}\+3\mathrm{dy}\right]\right)\right)$

$\left[\begin{array}{c}\mathrm{\_DG}\left(\left[\left[''form''\,M\,1\right]\,\left[\left[\left[1\right]\,1\right]\,\left[\left[2\right]\,2\right]\right]\right]\right)\\ \mathrm{\_DG}\left(\left[\left[''form''\,M\,1\right]\,\left[\left[\left[1\right]\,1\right]\,\left[\left[2\right]\,3\right]\right]\right]\right)\end{array}\right]$

$C≔\mathrm{Matrix}\left(\left[\left[1\,2\right]\,\left[3\,4\right]\right]\right)$

$\left[\begin{array}{rr}1& 2\\ 3& 4\end{array}\right]$

$E≔\mathrm{LinearAlgebra}:-\mathrm{Transpose}\left(A\right)$

$\left[\begin{array}{cc}\mathrm{\_DG}\left(\left[\left[''form''\,M\,1\right]\,\left[\left[\left[1\right]\,1\right]\,\left[\left[2\right]\,-1\right]\right]\right]\right)& \mathrm{\_DG}\left(\left[\left[''form''\,M\,1\right]\,\left[\left[\left[1\right]\,1\right]\,\left[\left[2\right]\,1\right]\right]\right]\right)\end{array}\right]$

$F≔\mathrm{Matrix}\left(\mathrm{evalDG}\left(\left[\left[\mathrm{dx}-\mathrm{dz}\,\mathrm{dy}\right]\,\left[\mathrm{dz}\,\mathrm{dx}\+\mathrm{dy}\+3\mathrm{dz}\right]\right]\right)\right)$

$\left[\begin{array}{cc}\mathrm{\_DG}\left(\left[\left[''form''\,M\,1\right]\,\left[\left[\left[1\right]\,1\right]\,\left[\left[3\right]\,-1\right]\right]\right]\right)& \mathrm{\_DG}\left(\left[\left[''form''\,M\,1\right]\,\left[\left[\left[2\right]\,1\right]\right]\right]\right)\\ \mathrm{\_DG}\left(\left[\left[''form''\,M\,1\right]\,\left[\left[\left[3\right]\,1\right]\right]\right]\right)& \mathrm{\_DG}\left(\left[\left[''form''\,M\,1\right]\,\left[\left[\left[1\right]\,1\right]\,\left[\left[2\right]\,1\right]\,\left[\left[3\right]\,3\right]\right]\right]\right)\end{array}\right]$

$A\&MatrixPlusB$

$\left[\begin{array}{c}\mathrm{\_DG}\left(\left[\left[''form''\,M\,1\right]\,\left[\left[\left[1\right]\,2\right]\,\left[\left[2\right]\,1\right]\right]\right]\right)\\ \mathrm{\_DG}\left(\left[\left[''form''\,M\,1\right]\,\left[\left[\left[1\right]\,2\right]\,\left[\left[2\right]\,4\right]\right]\right]\right)\end{array}\right]$

$A\&MatrixMinusB$

$\left[\begin{array}{c}\mathrm{\_DG}\left(\left[\left[''form''\,M\,1\right]\,\left[\left[\left[2\right]\,-3\right]\right]\right]\right)\\ \mathrm{\_DG}\left(\left[\left[''form''\,M\,1\right]\,\left[\left[\left[2\right]\,-2\right]\right]\right]\right)\end{array}\right]$

$a\&MatrixMultA$

$\left[\begin{array}{c}\mathrm{\_DG}\left(\left[\left[''form''\,M\,1\right]\,\left[\left[\left[1\right]\,a\right]\,\left[\left[2\right]\,-a\right]\right]\right]\right)\\ \mathrm{\_DG}\left(\left[\left[''form''\,M\,1\right]\,\left[\left[\left[1\right]\,a\right]\,\left[\left[2\right]\,a\right]\right]\right]\right)\end{array}\right]$

$C\&MatrixMultA$

$\left[\begin{array}{c}\mathrm{\_DG}\left(\left[\left[''form''\,M\,1\right]\,\left[\left[\left[1\right]\,3\right]\,\left[\left[2\right]\,1\right]\right]\right]\right)\\ \mathrm{\_DG}\left(\left[\left[''form''\,M\,1\right]\,\left[\left[\left[1\right]\,7\right]\,\left[\left[2\right]\,1\right]\right]\right]\right)\end{array}\right]$

$E\&MatrixMultC$

$\left[\begin{array}{cc}\mathrm{\_DG}\left(\left[\left[''form''\,M\,1\right]\,\left[\left[\left[1\right]\,4\right]\,\left[\left[2\right]\,2\right]\right]\right]\right)& \mathrm{\_DG}\left(\left[\left[''form''\,M\,1\right]\,\left[\left[\left[1\right]\,6\right]\,\left[\left[2\right]\,2\right]\right]\right]\right)\end{array}\right]$

$E\&MatrixWedgeB$

$\left[\begin{array}{c}\mathrm{\_DG}\left(\left[\left[''form''\,M\,2\right]\,\left[\left[\left[1\,2\right]\,5\right]\right]\right]\right)\end{array}\right]$

$C\&MatrixMultF$

$\left[\begin{array}{cc}\mathrm{\_DG}\left(\left[\left[''form''\,M\,1\right]\,\left[\left[\left[1\right]\,1\right]\,\left[\left[3\right]\,1\right]\right]\right]\right)& \mathrm{\_DG}\left(\left[\left[''form''\,M\,1\right]\,\left[\left[\left[1\right]\,2\right]\,\left[\left[2\right]\,3\right]\,\left[\left[3\right]\,6\right]\right]\right]\right)\\ \mathrm{\_DG}\left(\left[\left[''form''\,M\,1\right]\,\left[\left[\left[1\right]\,3\right]\,\left[\left[3\right]\,1\right]\right]\right]\right)& \mathrm{\_DG}\left(\left[\left[''form''\,M\,1\right]\,\left[\left[\left[1\right]\,4\right]\,\left[\left[2\right]\,7\right]\,\left[\left[3\right]\,12\right]\right]\right]\right)\end{array}\right]$

$F\&MatrixWedgeA$

$\left[\begin{array}{c}\mathrm{\_DG}\left(\left[\left[''form''\,M\,2\right]\,\left[\left[\left[1\,2\right]\,-2\right]\,\left[\left[1\,3\right]\,1\right]\,\left[\left[2\,3\right]\,-1\right]\right]\right]\right)\\ \mathrm{\_DG}\left(\left[\left[''form''\,M\,2\right]\,\left[\left[\left[1\,3\right]\,-4\right]\,\left[\left[2\,3\right]\,-2\right]\right]\right]\right)\end{array}\right]$

$F\&MatrixWedgeF$

$\left[\begin{array}{cc}\mathrm{\_DG}\left(\left[\left[''form''\,M\,2\right]\,\left[\left[\left[2\,3\right]\,1\right]\right]\right]\right)& \mathrm{\_DG}\left(\left[\left[''form''\,M\,2\right]\,\left[\left[\left[2\,3\right]\,4\right]\right]\right]\right)\\ \mathrm{\_DG}\left(\left[\left[''form''\,M\,2\right]\,\left[\left[\left[2\,3\right]\,1\right]\right]\right]\right)& \mathrm{\_DG}\left(\left[\left[''form''\,M\,2\right]\,\left[\left[\left[2\,3\right]\,-1\right]\right]\right]\right)\end{array}\right]$