OpenMath Content Dictionary: linalg6

Canonical URL:

http://www.openmath.org/cd/linalg6.ocd

CD Base:

http://www.openmath.org/cd

CD File:

linalg6.ocd

CD as XML Encoded OpenMath:

linalg6.omcd

Defines:

matrix_tensor, vector_tensor

Date:

1999-07-15

Version:

2

Review Date:

2003-04-01

Status:

experimental

Uses CD:

alg1, arith1, linalg1, nums1, logic1, relation1

This CD contains symbols for the construction of some new matrices from old ones.

matrix_tensor

Description:

This symbol denotes a n-nary function which is used to construct the tensor product matrix of its arguments, which must be matrices.

Commented Mathematical property (CMP):

If A is an m x r matrix B is an n x s matrix, then their Kronecker product is the m n x rs matrix whose i + (k-1) m, j + (l-1) n entry is A_{i,j} B_{k,l},

Formal Mathematical property (FMP):

OpenMath XML (source)

<OMOBJ xmlns="http://www.openmath.org/OpenMath" version="2.0"> 
<OMA><OMS cd="logic1" name="implies"/>
     <OMA><OMS cd="logic1" name="and"/>
          <OMA><OMS cd="relation1" name="eq"/>
               <OMA><OMS cd="linalg4" name="rowcount"/>
                    <OMV name="A"/>
               </OMA>
               <OMV name="m"/>
          </OMA>
          <OMA><OMS cd="relation1" name="eq"/>
               <OMA><OMS cd="linalg4" name="columncount"/>
                    <OMV name="A"/>
               </OMA>
               <OMV name="n"/>
          </OMA>
     </OMA>
     <OMA><OMS cd="relation1" name="eq"/>
          <OMA><OMS cd="linalg1" name="matrix_selector"/>
               <OMA><OMS cd="linalg6" name="matrix_tensor"/>
                    <OMV name="A"/>         <OMV name="B"/>
               </OMA>
               <OMA><OMS cd="arith1" name="plus"/>
                    <OMV name="i"/>
                    <OMA><OMS cd="arith1" name="times"/>
                         <OMA><OMS cd="arith1" name="minus"/>
                              <OMV name="k"/><OMI>1</OMI>
                         </OMA>
                         <OMV name="m"/>
                    </OMA>
               </OMA>
               <OMA><OMS cd="arith1" name="plus"/>
                    <OMV name="j"/>
                    <OMA><OMS cd="arith1" name="times"/>
                         <OMA><OMS cd="arith1" name="minus"/>
                              <OMV name="l"/><OMI>1</OMI>
                         </OMA>
                         <OMV name="n"/>
                    </OMA>
               </OMA>
          </OMA>
          <OMA><OMS cd="arith1" name="times"/>
               <OMA><OMS cd="linalg1" name="matrix_selector"/>
                    <OMV name="A"/>
                    <OMV name="i"/><OMV name="j"/>
               </OMA>
               <OMA><OMS cd="linalg1" name="matrix_selector"/>
                    <OMV name="B"/>
                    <OMV name="k"/><OMV name="l"/>
               </OMA>
          </OMA>
     </OMA>
</OMA>
 </OMOBJ>

Strict Content MathML

<math xmlns="http://www.w3.org/1998/Math/MathML">
 <apply><csymbol cd="logic1">implies</csymbol>
  <apply><csymbol cd="logic1">and</csymbol>
   <apply><csymbol cd="relation1">eq</csymbol>
    <apply><csymbol cd="linalg4">rowcount</csymbol><ci>A</ci></apply>
    <ci>m</ci>
   </apply>
   <apply><csymbol cd="relation1">eq</csymbol>
    <apply><csymbol cd="linalg4">columncount</csymbol><ci>A</ci></apply>
    <ci>n</ci>
   </apply>
  </apply>
  <apply><csymbol cd="relation1">eq</csymbol>
   <apply><csymbol cd="linalg1">matrix_selector</csymbol>
    <apply><csymbol cd="linalg6">matrix_tensor</csymbol><ci>A</ci><ci>B</ci></apply>
    <apply><csymbol cd="arith1">plus</csymbol>
     <ci>i</ci>
     <apply><csymbol cd="arith1">times</csymbol>
      <apply><csymbol cd="arith1">minus</csymbol><ci>k</ci><cn>1</cn></apply>
      <ci>m</ci>
     </apply>
    </apply>
    <apply><csymbol cd="arith1">plus</csymbol>
     <ci>j</ci>
     <apply><csymbol cd="arith1">times</csymbol>
      <apply><csymbol cd="arith1">minus</csymbol><ci>l</ci><cn>1</cn></apply>
      <ci>n</ci>
     </apply>
    </apply>
   </apply>
   <apply><csymbol cd="arith1">times</csymbol>
    <apply><csymbol cd="linalg1">matrix_selector</csymbol><ci>A</ci><ci>i</ci><ci>j</ci></apply>
    <apply><csymbol cd="linalg1">matrix_selector</csymbol><ci>B</ci><ci>k</ci><ci>l</ci></apply>
   </apply>
  </apply>
 </apply>
</math>

Prefix

implies (and (eq (rowcount ( A) , m) , eq (columncount ( A) , n) ) , eq (matrix_selector (matrix_tensor ( A, B) , plus ( i, times (minus ( k, 1) , m) ) , plus ( j, times (minus ( l, 1) , n) ) ) , times (matrix_selector ( A, i, j) , matrix_selector ( B, k, l) ) ) )

Popcorn

linalg4.rowcount($A) = $m and linalg4.columncount($A) = $n ==> linalg1.matrix_selector(linalg6.matrix_tensor($A, $B), $i + ($k - 1) * $m, $j + ($l - 1) * $n) = linalg1.matrix_selector($A, $i, $j) * linalg1.matrix_selector($B, $k, $l)

Rendered Presentation MathML

$$\mathrm{rowcount}\left(A\right)=m\wedge \mathrm{columncount}\left(A\right)=n\Rightarrow {j+(l-1)n}_{\left(\mathrm{matrix\_tensor}(A,B),i+(k-1)m\right)}=j_{\left(A,i\right)}{l}_{\left(B,k\right)}$$

Signatures:

sts

vector_tensor

Description:

This symbol denotes a n-nary function which is used to construct the tensor product vector of its arguments, which must be vectors.

Signatures:

sts