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MathGroup Archive 1998

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Re: tensor/matrix calculations


  • To: mathgroup@smc.vnet.net
  • Subject: [mg10359] Re: tensor/matrix calculations
  • From: Paul Abbott <paul@physics.uwa.edu.au>
  • Date: Mon, 12 Jan 1998 04:09:55 -0500
  • Organization: University of Western Australia
  • References: <694bpr$mqt@smc.vnet.net>

Jan-Willem Roorda wrote:

> How can I calculate the standard matrix product of  two
> matrixrepresentations of a tensor. The dot . produkt does'nt work.
> 
> Jan-Willem Roorda

For the Kronecker product you need to use Outer.  Here is a simple
example:

In[1]:= << "LinearAlgebra`MatrixManipulation`" In[2]:= A = Table[a[i,
j], {i, 2}, {j, 2}]; In[3]:= B = Table[b[i, j], {i, 2}, {j, 2}];
In[4]:= BlockMatrix[Outer[Times, A, B]] Out[4]=
{{a[1, 1] b[1, 1], a[1, 1] b[1, 2], a[1, 2] b[1, 1], 
 
   a[1, 2] b[1, 2]}, {a[1, 1] b[2, 1], a[1, 1] b[2, 2], 
 
   a[1, 2] b[2, 1], a[1, 2] b[2, 2]}, 
 
  {a[2, 1] b[1, 1], a[2, 1] b[1, 2], a[2, 2] b[1, 1], 
 
   a[2, 2] b[1, 2]}, {a[2, 1] b[2, 1], a[2, 1] b[2, 2], 
 
   a[2, 2] b[2, 1], a[2, 2] b[2, 2]}}

This was discussed in The Mathematica Journal 2(2):36-37 and 2(4):38. I
have appended a Notebook below which gives a better-formatted example
(using CircleTimes).

____________________________________________________________________ 
Paul Abbott                                   Phone: +61-8-9380-2734
Department of Physics                           Fax: +61-8-9380-1014
The University of Western Australia            Nedlands WA  6907       
mailto:paul@physics.uwa.edu.au  AUSTRALIA                            
http://www.pd.uwa.edu.au/~paul

            God IS a weakly left-handed dice player
____________________________________________________________________

Notebook[{

Cell[CellGroupData[{
Cell["Kronecker Products", "Section"],

Cell[TextData[{
  StyleBox["TMJ ",
    FontSlant->"Italic"],
  StyleBox["2",
    FontWeight->"Bold"],
  "(2):36-37, ",
  StyleBox["2",
    FontWeight->"Bold"],
  "(4):38 (addendum), ",
  StyleBox["1",
    FontWeight->"Bold"],
  "(3):59 (Distance matrices), Lewis Stiller" }], "Text"],

Cell[TextData[{
  "The symbol \[CircleTimes] represents an ",
  "infix",
  " (and ",
  "prefix",
  ") operator:"
}], "Text",
  CellTags->"CircleTimes"],

Cell[CellGroupData[{

Cell[BoxData[
    \(\(?\[CircleTimes]\)\)], "Input"],

Cell[BoxData[
    \("An infix and prefix operator. \!\(x \[CircleTimes] y\) is by
default \
interpreted as \!\(CircleTimes[x, y]\) ; \!\(\[CircleTimes] x\) is by
default \
interpreted as \!\(CircleTimes[x]\)."\)], "Print"] }, Open  ]],

Cell["However, there are no built-in rules for CircleTimes:", "Text"],

Cell[CellGroupData[{

Cell[BoxData[
    \(\(?CircleTimes\)\)], "Input"],

Cell[BoxData[
    \(Information::"notfound" \( : \ \) 
      "Symbol \!\(\"CircleTimes\"\) not found."\)], "Message"] }, Open 
]],

Cell["\<\
Here is simple implementation of the matrix Kronecker product:\ \>",
   "Text",
  CellTags->{
  "Kronecker product", "Direct product",
"LinearAlgebra`MatrixManipulation`", 
    "BlockMatrix", "Outer"}],

Cell[BoxData[
    \(TraditionalForm\`<< LinearAlgebra`MatrixManipulation`\)],
"Input"],

Cell[BoxData[
    \(TraditionalForm
    \`a_\[CircleTimes]b_ := BlockMatrix[Outer[Times, a, b]]\)],
"Input"],

Cell[BoxData[
    \(TraditionalForm
    \`a_\[CircleTimes]b_ := BlockMatrix(Outer(Times, a, b))\)],
"Input"],

Cell["For example,", "Text"],

Cell[CellGroupData[{

Cell[BoxData[
    \(TraditionalForm
    \`\[ScriptCapitalA] = Table[\[ScriptA]\_\(i, j\), {i, 2}, {j,
2}]\)], 
  "Input"],

Cell[BoxData[
    FormBox[
      RowBox[{"(", GridBox[{
            {\(\[ScriptA]\_\(1, 1\)\), \(\[ScriptA]\_\(1, 2\)\)},
            {\(\[ScriptA]\_\(2, 1\)\), \(\[ScriptA]\_\(2, 2\)\)}
            }], ")"}], TraditionalForm]], "Output"] }, Open  ]],

Cell[CellGroupData[{

Cell[BoxData[
    \(TraditionalForm
    \`\[ScriptCapitalB] = Table[\[ScriptB]\_\(i, j\), {i, 2}, {j,
2}]\)], 
  "Input"],

Cell[BoxData[
    FormBox[
      RowBox[{"(", GridBox[{
            {\(\[ScriptB]\_\(1, 1\)\), \(\[ScriptB]\_\(1, 2\)\)},
            {\(\[ScriptB]\_\(2, 1\)\), \(\[ScriptB]\_\(2, 2\)\)}
            }], ")"}], TraditionalForm]], "Output"] }, Open  ]],

Cell[CellGroupData[{

Cell[BoxData[
    \(TraditionalForm\`Outer[Times, \[ScriptCapitalA],
\[ScriptCapitalB]]\)], 
  "Input"],

Cell[BoxData[
    FormBox[
      InterpretationBox[
        RowBox[{"(", GridBox[{
              {
                RowBox[{"(", GridBox[{
                      {\(\[ScriptA]\_\(1, 1\)\ \[ScriptB]\_\(1, 1\)\), 
                        \(\[ScriptA]\_\(1, 1\)\ \[ScriptB]\_\(1, 2\)\)},
                      {\(\[ScriptA]\_\(1, 1\)\ \[ScriptB]\_\(2, 1\)\), 
                        \(\[ScriptA]\_\(1, 1\)\ \[ScriptB]\_\(2, 2\)\)}
                      }], ")"}], 
                RowBox[{"(", GridBox[{
                      {\(\[ScriptA]\_\(1, 2\)\ \[ScriptB]\_\(1, 1\)\), 
                        \(\[ScriptA]\_\(1, 2\)\ \[ScriptB]\_\(1, 2\)\)},
                      {\(\[ScriptA]\_\(1, 2\)\ \[ScriptB]\_\(2, 1\)\), 
                        \(\[ScriptA]\_\(1, 2\)\ \[ScriptB]\_\(2, 2\)\)}
                      }], ")"}]},
              {
                RowBox[{"(", GridBox[{
                      {\(\[ScriptA]\_\(2, 1\)\ \[ScriptB]\_\(1, 1\)\), 
                        \(\[ScriptA]\_\(2, 1\)\ \[ScriptB]\_\(1, 2\)\)},
                      {\(\[ScriptA]\_\(2, 1\)\ \[ScriptB]\_\(2, 1\)\), 
                        \(\[ScriptA]\_\(2, 1\)\ \[ScriptB]\_\(2, 2\)\)}
                      }], ")"}], 
                RowBox[{"(", GridBox[{
                      {\(\[ScriptA]\_\(2, 2\)\ \[ScriptB]\_\(1, 1\)\), 
                        \(\[ScriptA]\_\(2, 2\)\ \[ScriptB]\_\(1, 2\)\)},
                      {\(\[ScriptA]\_\(2, 2\)\ \[ScriptB]\_\(2, 1\)\), 
                        \(\[ScriptA]\_\(2, 2\)\ \[ScriptB]\_\(2, 2\)\)}
                      }], ")"}]}
              }], ")"}],
        MatrixForm[ {{{{
          Times[ 
            Subscript[ \[ScriptA], 1, 1], 
            Subscript[ \[ScriptB], 1, 1]], 
          Times[ 
            Subscript[ \[ScriptA], 1, 1], 
            Subscript[ \[ScriptB], 1, 2]]}, {
          Times[ 
            Subscript[ \[ScriptA], 1, 1], 
            Subscript[ \[ScriptB], 2, 1]], 
          Times[ 
            Subscript[ \[ScriptA], 1, 1], 
            Subscript[ \[ScriptB], 2, 2]]}}, {{
          Times[ 
            Subscript[ \[ScriptA], 1, 2], 
            Subscript[ \[ScriptB], 1, 1]], 
          Times[ 
            Subscript[ \[ScriptA], 1, 2], 
            Subscript[ \[ScriptB], 1, 2]]}, {
          Times[ 
            Subscript[ \[ScriptA], 1, 2], 
            Subscript[ \[ScriptB], 2, 1]], 
          Times[ 
            Subscript[ \[ScriptA], 1, 2], 
            Subscript[ \[ScriptB], 2, 2]]}}}, {{{
          Times[ 
            Subscript[ \[ScriptA], 2, 1], 
            Subscript[ \[ScriptB], 1, 1]], 
          Times[ 
            Subscript[ \[ScriptA], 2, 1], 
            Subscript[ \[ScriptB], 1, 2]]}, {
          Times[ 
            Subscript[ \[ScriptA], 2, 1], 
            Subscript[ \[ScriptB], 2, 1]], 
          Times[ 
            Subscript[ \[ScriptA], 2, 1], 
            Subscript[ \[ScriptB], 2, 2]]}}, {{
          Times[ 
            Subscript[ \[ScriptA], 2, 2], 
            Subscript[ \[ScriptB], 1, 1]], 
          Times[ 
            Subscript[ \[ScriptA], 2, 2], 
            Subscript[ \[ScriptB], 1, 2]]}, {
          Times[ 
            Subscript[ \[ScriptA], 2, 2], 
            Subscript[ \[ScriptB], 2, 1]], 
          Times[ 
            Subscript[ \[ScriptA], 2, 2], 
            Subscript[ \[ScriptB], 2, 2]]}}}}, TableDepth -> 2]], 
      TraditionalForm]], "Output"]
}, Open  ]],

Cell[CellGroupData[{

Cell[BoxData[
    \(TraditionalForm
    \`\[ScriptCapitalA]\[ScriptCapitalB] = 
      \[ScriptCapitalA]\[CircleTimes]\[ScriptCapitalB]\)], "Input"],

Cell[BoxData[
    FormBox[
      RowBox[{"(", GridBox[{
            {\(\[ScriptA]\_\(1, 1\)\ \[ScriptB]\_\(1, 1\)\), 
              \(\[ScriptA]\_\(1, 1\)\ \[ScriptB]\_\(1, 2\)\), 
              \(\[ScriptA]\_\(1, 2\)\ \[ScriptB]\_\(1, 1\)\), 
              \(\[ScriptA]\_\(1, 2\)\ \[ScriptB]\_\(1, 2\)\)},
            {\(\[ScriptA]\_\(1, 1\)\ \[ScriptB]\_\(2, 1\)\), 
              \(\[ScriptA]\_\(1, 1\)\ \[ScriptB]\_\(2, 2\)\), 
              \(\[ScriptA]\_\(1, 2\)\ \[ScriptB]\_\(2, 1\)\), 
              \(\[ScriptA]\_\(1, 2\)\ \[ScriptB]\_\(2, 2\)\)},
            {\(\[ScriptA]\_\(2, 1\)\ \[ScriptB]\_\(1, 1\)\), 
              \(\[ScriptA]\_\(2, 1\)\ \[ScriptB]\_\(1, 2\)\), 
              \(\[ScriptA]\_\(2, 2\)\ \[ScriptB]\_\(1, 1\)\), 
              \(\[ScriptA]\_\(2, 2\)\ \[ScriptB]\_\(1, 2\)\)},
            {\(\[ScriptA]\_\(2, 1\)\ \[ScriptB]\_\(2, 1\)\), 
              \(\[ScriptA]\_\(2, 1\)\ \[ScriptB]\_\(2, 2\)\), 
              \(\[ScriptA]\_\(2, 2\)\ \[ScriptB]\_\(2, 1\)\), 
              \(\[ScriptA]\_\(2, 2\)\ \[ScriptB]\_\(2, 2\)\)}
            }], ")"}], TraditionalForm]], "Output"] }, Open  ]]
}, Open  ]]
}
]



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