&MatrixMinus - Maple Help

DifferentialGeometry:-Tools[&MatrixMinus, &MatrixMult, &MatrixPlus, &MatrixWedge]

 Calling Sequence A &MatrixMinus B - subtract two Matrices/Vectors of vectors, differential forms or tensors A &MatrixMult C - multiply a Matrix/Vector A of vectors, differential forms or tensors by a scalar C or a Matrix/Vector C of scalars C &MatrixMult A - multiply a Matrix A of vectors, differential forms or tensors by a scalar C or a Matrix/Vector C of scalars A &MatrixPlus B - add two Matrices/Vectors of vectors, differential forms or tensors E &MatrixWedge F - calculate the Matrix wedge product of two Matrices/Vectors of differential forms.

Parameters

 A, B - two Matrices/Vectors of vectors, differential forms or tensors C - a scalar or a Matrix/Vector of scalars E, F - two Matrices/Vectors of differential forms

Description

 • 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.

Examples

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

Define a 3-dimensional manifold M with coordinates [x, y, z].

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

Example 1

Define two column Vectors of 1 forms A, B; a 2x2 matrix C of scalars; a row Vector of 1 forms E and a 2x2 Matrix of 1 forms F.

 > $A≔\mathrm{Vector}\left(\mathrm{evalDG}\left(\left[\mathrm{dx}-\mathrm{dy},\mathrm{dy}+\mathrm{dx}\right]\right)\right)$
 $\left[\begin{array}{c}\mathrm{_DG}{}\left(\left[\left[{"form"},M,1\right],\left[\left[\left[1\right],1\right],\left[\left[2\right],-1\right]\right]\right]\right)\\ \mathrm{_DG}{}\left(\left[\left[{"form"},M,1\right],\left[\left[\left[1\right],1\right],\left[\left[2\right],1\right]\right]\right]\right)\end{array}\right]$ (1)
 > $B≔\mathrm{Vector}\left(\mathrm{evalDG}\left(\left[\mathrm{dx}+2\mathrm{dy},\mathrm{dx}+3\mathrm{dy}\right]\right)\right)$
 $\left[\begin{array}{c}\mathrm{_DG}{}\left(\left[\left[{"form"},M,1\right],\left[\left[\left[1\right],1\right],\left[\left[2\right],2\right]\right]\right]\right)\\ \mathrm{_DG}{}\left(\left[\left[{"form"},M,1\right],\left[\left[\left[1\right],1\right],\left[\left[2\right],3\right]\right]\right]\right)\end{array}\right]$ (2)
 > $C≔\mathrm{Matrix}\left(\left[\left[1,2\right],\left[3,4\right]\right]\right)$
 $\left[\begin{array}{rr}1& 2\\ 3& 4\end{array}\right]$ (3)
 > $E≔\mathrm{LinearAlgebra}:-\mathrm{Transpose}\left(A\right)$
 $\left[\begin{array}{cc}\mathrm{_DG}{}\left(\left[\left[{"form"},M,1\right],\left[\left[\left[1\right],1\right],\left[\left[2\right],-1\right]\right]\right]\right)& \mathrm{_DG}{}\left(\left[\left[{"form"},M,1\right],\left[\left[\left[1\right],1\right],\left[\left[2\right],1\right]\right]\right]\right)\end{array}\right]$ (4)
 > $F≔\mathrm{Matrix}\left(\mathrm{evalDG}\left(\left[\left[\mathrm{dx}-\mathrm{dz},\mathrm{dy}\right],\left[\mathrm{dz},\mathrm{dx}+\mathrm{dy}+3\mathrm{dz}\right]\right]\right)\right)$
 $\left[\begin{array}{cc}\mathrm{_DG}{}\left(\left[\left[{"form"},M,1\right],\left[\left[\left[1\right],1\right],\left[\left[3\right],-1\right]\right]\right]\right)& \mathrm{_DG}{}\left(\left[\left[{"form"},M,1\right],\left[\left[\left[2\right],1\right]\right]\right]\right)\\ \mathrm{_DG}{}\left(\left[\left[{"form"},M,1\right],\left[\left[\left[3\right],1\right]\right]\right]\right)& \mathrm{_DG}{}\left(\left[\left[{"form"},M,1\right],\left[\left[\left[1\right],1\right],\left[\left[2\right],1\right],\left[\left[3\right],3\right]\right]\right]\right)\end{array}\right]$ (5)

Perform various arithmetic operations with the quantities A, B, C, E, F.

 > $A&MatrixPlusB$
 $\left[\begin{array}{c}\mathrm{_DG}{}\left(\left[\left[{"form"},M,1\right],\left[\left[\left[1\right],2\right],\left[\left[2\right],1\right]\right]\right]\right)\\ \mathrm{_DG}{}\left(\left[\left[{"form"},M,1\right],\left[\left[\left[1\right],2\right],\left[\left[2\right],4\right]\right]\right]\right)\end{array}\right]$ (6)
 > $A&MatrixMinusB$
 $\left[\begin{array}{c}\mathrm{_DG}{}\left(\left[\left[{"form"},M,1\right],\left[\left[\left[2\right],-3\right]\right]\right]\right)\\ \mathrm{_DG}{}\left(\left[\left[{"form"},M,1\right],\left[\left[\left[2\right],-2\right]\right]\right]\right)\end{array}\right]$ (7)
 > $a&MatrixMultA$
 $\left[\begin{array}{c}\mathrm{_DG}{}\left(\left[\left[{"form"},M,1\right],\left[\left[\left[1\right],a\right],\left[\left[2\right],-a\right]\right]\right]\right)\\ \mathrm{_DG}{}\left(\left[\left[{"form"},M,1\right],\left[\left[\left[1\right],a\right],\left[\left[2\right],a\right]\right]\right]\right)\end{array}\right]$ (8)
 > $C&MatrixMultA$
 $\left[\begin{array}{c}\mathrm{_DG}{}\left(\left[\left[{"form"},M,1\right],\left[\left[\left[1\right],3\right],\left[\left[2\right],1\right]\right]\right]\right)\\ \mathrm{_DG}{}\left(\left[\left[{"form"},M,1\right],\left[\left[\left[1\right],7\right],\left[\left[2\right],1\right]\right]\right]\right)\end{array}\right]$ (9)
 > $E&MatrixMultC$
 $\left[\begin{array}{cc}\mathrm{_DG}{}\left(\left[\left[{"form"},M,1\right],\left[\left[\left[1\right],4\right],\left[\left[2\right],2\right]\right]\right]\right)& \mathrm{_DG}{}\left(\left[\left[{"form"},M,1\right],\left[\left[\left[1\right],6\right],\left[\left[2\right],2\right]\right]\right]\right)\end{array}\right]$ (10)
 > $E&MatrixWedgeB$
 $\left[\begin{array}{c}\mathrm{_DG}{}\left(\left[\left[{"form"},M,2\right],\left[\left[\left[1,2\right],5\right]\right]\right]\right)\end{array}\right]$ (11)
 > $C&MatrixMultF$
 $\left[\begin{array}{cc}\mathrm{_DG}{}\left(\left[\left[{"form"},M,1\right],\left[\left[\left[1\right],1\right],\left[\left[3\right],1\right]\right]\right]\right)& \mathrm{_DG}{}\left(\left[\left[{"form"},M,1\right],\left[\left[\left[1\right],2\right],\left[\left[2\right],3\right],\left[\left[3\right],6\right]\right]\right]\right)\\ \mathrm{_DG}{}\left(\left[\left[{"form"},M,1\right],\left[\left[\left[1\right],3\right],\left[\left[3\right],1\right]\right]\right]\right)& \mathrm{_DG}{}\left(\left[\left[{"form"},M,1\right],\left[\left[\left[1\right],4\right],\left[\left[2\right],7\right],\left[\left[3\right],12\right]\right]\right]\right)\end{array}\right]$ (12)
 > $F&MatrixWedgeA$
 $\left[\begin{array}{c}\mathrm{_DG}{}\left(\left[\left[{"form"},M,2\right],\left[\left[\left[1,2\right],-2\right],\left[\left[1,3\right],1\right],\left[\left[2,3\right],-1\right]\right]\right]\right)\\ \mathrm{_DG}{}\left(\left[\left[{"form"},M,2\right],\left[\left[\left[1,3\right],-4\right],\left[\left[2,3\right],-2\right]\right]\right]\right)\end{array}\right]$ (13)
 > $F&MatrixWedgeF$
 $\left[\begin{array}{cc}\mathrm{_DG}{}\left(\left[\left[{"form"},M,2\right],\left[\left[\left[2,3\right],1\right]\right]\right]\right)& \mathrm{_DG}{}\left(\left[\left[{"form"},M,2\right],\left[\left[\left[2,3\right],4\right]\right]\right]\right)\\ \mathrm{_DG}{}\left(\left[\left[{"form"},M,2\right],\left[\left[\left[2,3\right],1\right]\right]\right]\right)& \mathrm{_DG}{}\left(\left[\left[{"form"},M,2\right],\left[\left[\left[2,3\right],-1\right]\right]\right]\right)\end{array}\right]$ (14)