OpenMath Content Dictionary: linalgrank1

Canonical URL:

http://www.openmath.org/cd/linalgrank1.ocd

CD Base:

http://www.openmath.org/cd

CD File:

linalgrank1.ocd

CD as XML Encoded OpenMath:

linalgrank1.omcd

Defines:

dual_kernel_matrix, kernel_matrix, rank

Date:

2004-11-30

Version:

3

Review Date:

2006-03-30

Status:

experimental

---

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     holder permits you to develop any derived work from this document
     provided that the following conditions are met.
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          this document, and maintains a prominent reference in the 
          work to the original source.
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          both this document and the derived work are Content Dictionaries
          then the derived work must include a different CDName element,
          chosen so that it cannot be confused with any works adopted by
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This CD defines symbols for basic linear algebra (over a division ring) related to rank and kernel.

It is constructed by A.M. Cohen, who took the definition of rank from a former version of linalg4.ocd.

---

rank

Role:

application

Description:

This symbol represents the function which takes one matrix argument and returns the number of linearly independent rows (or columns) of that matrix.

Commented Mathematical property (CMP):

The rank of the n x n identity matrix is equal to n.

Formal Mathematical property (FMP):

OpenMath XML (source)

<OMOBJ xmlns="http://www.openmath.org/OpenMath" version="2.0" cdbase="http://www.openmath.org/cd">
  <OMA><OMS cd="relation1" name="eq"/>
       <OMA><OMS cd="linalgrank1" name="rank"/>
            <OMA><OMS cd="linalg4mat" name="identity"/>
                 <OMV name="n"/>
            </OMA>
       </OMA>
       <OMV name="n"/>
  </OMA>
</OMOBJ>

Strict Content MathML

<math xmlns="http://www.w3.org/1998/Math/MathML">
 <apply><csymbol cd="relation1">eq</csymbol>
  <apply><csymbol cd="linalgrank1">rank</csymbol>
   <apply><csymbol cd="linalg4mat">identity</csymbol><ci>n</ci></apply>
  </apply>
  <ci>n</ci>
 </apply>
</math>

Prefix

eq (rank (identity ( n) ) , n)

Popcorn

linalgrank1.rank(linalg4mat.identity($n)) = $n

Rendered Presentation MathML

$$\mathrm{rank}\left(\mathrm{identity}\left(n\right)\right)=n$$

Signatures:

sts

---

[Next: kernel_matrix] [Last: dual_kernel_matrix] [Top]

---

kernel_matrix

Role:

application

Description:

This symbol represents a unary function whose argument should be a matrix. When applied to a matrix A, it represents a matrix whose rows are a basis of the kernel of A acting on the right.

Commented Mathematical property (CMP):

rowcount(kernel_matrix(A)) = rowcount(A) - rank(A)

Formal Mathematical property (FMP):

OpenMath XML (source)

<OMOBJ xmlns="http://www.openmath.org/OpenMath" version="2.0" cdbase="http://www.openmath.org/cd">
  <OMA><OMS cd="relation1" name="eq"/>
       <OMA><OMS cd="linalg3" name="rowcount"/>
            <OMA><OMS cd="linalgrank1" name="kernel_matrix"/>
                 <OMV name="A"/>
            </OMA>
      </OMA>      
      <OMA><OMS cd="arith1" name="minus"/>
           <OMA><OMS cd="linalg3" name="rowcount"/>
                <OMV name="A"/>
           </OMA>
           <OMA><OMS cd="linalgrank1" name="rank"/>
                <OMV name="A"/>
           </OMA>
      </OMA>
 </OMA>
</OMOBJ>

Strict Content MathML

<math xmlns="http://www.w3.org/1998/Math/MathML">
 <apply><csymbol cd="relation1">eq</csymbol>
  <apply><csymbol cd="linalg3">rowcount</csymbol>
   <apply><csymbol cd="linalgrank1">kernel_matrix</csymbol><ci>A</ci></apply>
  </apply>
  <apply><csymbol cd="arith1">minus</csymbol>
   <apply><csymbol cd="linalg3">rowcount</csymbol><ci>A</ci></apply>
   <apply><csymbol cd="linalgrank1">rank</csymbol><ci>A</ci></apply>
  </apply>
 </apply>
</math>

Prefix

eq (rowcount (kernel_matrix ( A) ) , minus (rowcount ( A) , rank ( A) ) )

Popcorn

linalg3.rowcount(linalgrank1.kernel_matrix($A)) = linalg3.rowcount($A) - linalgrank1.rank($A)

Rendered Presentation MathML

$$\mathrm{rowcount}\left(\mathrm{kernel\_matrix}\left(A\right)\right)=\mathrm{rowcount}\left(A\right)-\mathrm{rank}\left(A\right)$$

Commented Mathematical property (CMP):

kernel_matrix(A) * A = 0

Formal Mathematical property (FMP):

OpenMath XML (source)

<OMOBJ xmlns="http://www.openmath.org/OpenMath" version="2.0" cdbase="http://www.openmath.org/cd">
  <OMA><OMS cd="relation1" name="eq"/>
       <OMA><OMS cd="arith1" name="times"/>
            <OMA><OMS cd="linalgrank1" name="kernel_matrix"/>
                 <OMV name="A"/>
            </OMA>
            <OMV name="A"/>
      </OMA>      
      <OMA><OMS cd="linalg4mat" name="zero"/>
           <OMA><OMS cd="arith1" name="minus"/>
                <OMA><OMS cd="linalg3" name="rowcount"/>
                     <OMV name="A"/>
                </OMA>
                <OMA><OMS cd="linalgrank1" name="rank"/>
                     <OMV name="A"/>
                </OMA>
           </OMA>
           <OMA><OMS cd="linalg3" name="columncount"/>
                <OMV name="A"/>
           </OMA>
      </OMA>
 </OMA>
</OMOBJ>

Strict Content MathML

<math xmlns="http://www.w3.org/1998/Math/MathML">
 <apply><csymbol cd="relation1">eq</csymbol>
  <apply><csymbol cd="arith1">times</csymbol>
   <apply><csymbol cd="linalgrank1">kernel_matrix</csymbol><ci>A</ci></apply>
   <ci>A</ci>
  </apply>
  <apply><csymbol cd="linalg4mat">zero</csymbol>
   <apply><csymbol cd="arith1">minus</csymbol>
    <apply><csymbol cd="linalg3">rowcount</csymbol><ci>A</ci></apply>
    <apply><csymbol cd="linalgrank1">rank</csymbol><ci>A</ci></apply>
   </apply>
   <apply><csymbol cd="linalg3">columncount</csymbol><ci>A</ci></apply>
  </apply>
 </apply>
</math>

Prefix

eq (times (kernel_matrix ( A) , A) , zero (minus (rowcount ( A) , rank ( A) ) , columncount ( A) ) )

Popcorn

linalgrank1.kernel_matrix($A) * $A = linalg4mat.zero(linalg3.rowcount($A) - linalgrank1.rank($A), linalg3.columncount($A))

Rendered Presentation MathML

$$\mathrm{kernel\_matrix}\left(A\right)A=\mathrm{zero}(\mathrm{rowcount}\left(A\right)-\mathrm{rank}\left(A\right),\mathrm{columncount}\left(A\right))$$

Signatures:

sts

---

[Next: dual_kernel_matrix] [Previous: rank] [Top]

---

dual_kernel_matrix

Role:

application

Description:

This symbol represents a unary function whose argument should be a matrix. When applied to a matrix, it represents a list of column vectors spanning the kernel of the matrix acting on the left.

Commented Mathematical property (CMP):

columncount(kernel_matrix(A)) = columncount(A) - rank(A)

Formal Mathematical property (FMP):

OpenMath XML (source)

<OMOBJ xmlns="http://www.openmath.org/OpenMath" version="2.0" cdbase="http://www.openmath.org/cd">
  <OMA><OMS cd="relation1" name="eq"/>
       <OMA><OMS cd="linalg3" name="columncount"/>
            <OMA><OMS cd="linalgrank1" name="kernel_matrix"/>
                 <OMV name="A"/>
            </OMA>
      </OMA>      
      <OMA><OMS cd="arith1" name="minus"/>
           <OMA><OMS cd="linalg3" name="columncount"/>
                <OMV name="A"/>
           </OMA>
           <OMA><OMS cd="linalgrank1" name="rank"/>
                <OMV name="A"/>
           </OMA>
      </OMA>
 </OMA>
</OMOBJ>

Strict Content MathML

<math xmlns="http://www.w3.org/1998/Math/MathML">
 <apply><csymbol cd="relation1">eq</csymbol>
  <apply><csymbol cd="linalg3">columncount</csymbol>
   <apply><csymbol cd="linalgrank1">kernel_matrix</csymbol><ci>A</ci></apply>
  </apply>
  <apply><csymbol cd="arith1">minus</csymbol>
   <apply><csymbol cd="linalg3">columncount</csymbol><ci>A</ci></apply>
   <apply><csymbol cd="linalgrank1">rank</csymbol><ci>A</ci></apply>
  </apply>
 </apply>
</math>

Prefix

eq (columncount (kernel_matrix ( A) ) , minus (columncount ( A) , rank ( A) ) )

Popcorn

linalg3.columncount(linalgrank1.kernel_matrix($A)) = linalg3.columncount($A) - linalgrank1.rank($A)

Rendered Presentation MathML

$$\mathrm{columncount}\left(\mathrm{kernel\_matrix}\left(A\right)\right)=\mathrm{columncount}\left(A\right)-\mathrm{rank}\left(A\right)$$

Commented Mathematical property (CMP):

A * kernel_matrix(A) = 0

Formal Mathematical property (FMP):

OpenMath XML (source)

<OMOBJ xmlns="http://www.openmath.org/OpenMath" version="2.0" cdbase="http://www.openmath.org/cd">
  <OMA><OMS cd="relation1" name="eq"/>
       <OMA><OMS cd="arith1" name="times"/>
            <OMV name="A"/>
            <OMA><OMS cd="linalgrank1" name="kernel_matrix"/>
                 <OMV name="A"/>
            </OMA>
      </OMA>      
      <OMA><OMS cd="linalg4mat" name="zero"/>
           <OMA><OMS cd="linalg3" name="rowcount"/>
                <OMV name="A"/>
           </OMA>
           <OMA><OMS cd="arith1" name="minus"/>
                <OMA><OMS cd="linalg3" name="columncount"/>
                     <OMV name="A"/>
                </OMA>
                <OMA><OMS cd="linalgrank1" name="rank"/>
                     <OMV name="A"/>
                </OMA>
           </OMA>
      </OMA>
 </OMA>
</OMOBJ>

Strict Content MathML

<math xmlns="http://www.w3.org/1998/Math/MathML">
 <apply><csymbol cd="relation1">eq</csymbol>
  <apply><csymbol cd="arith1">times</csymbol>
   <ci>A</ci>
   <apply><csymbol cd="linalgrank1">kernel_matrix</csymbol><ci>A</ci></apply>
  </apply>
  <apply><csymbol cd="linalg4mat">zero</csymbol>
   <apply><csymbol cd="linalg3">rowcount</csymbol><ci>A</ci></apply>
   <apply><csymbol cd="arith1">minus</csymbol>
    <apply><csymbol cd="linalg3">columncount</csymbol><ci>A</ci></apply>
    <apply><csymbol cd="linalgrank1">rank</csymbol><ci>A</ci></apply>
   </apply>
  </apply>
 </apply>
</math>

Prefix

eq (times ( A, kernel_matrix ( A) ) , zero (rowcount ( A) , minus (columncount ( A) , rank ( A) ) ) )

Popcorn

$A * linalgrank1.kernel_matrix($A) = linalg4mat.zero(linalg3.rowcount($A), linalg3.columncount($A) - linalgrank1.rank($A))

Rendered Presentation MathML

$$A\mathrm{kernel\_matrix}\left(A\right)=\mathrm{zero}(\mathrm{rowcount}\left(A\right),\mathrm{columncount}\left(A\right)-\mathrm{rank}\left(A\right))$$

Signatures:

sts

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[First: rank] [Previous: kernel_matrix] [Top]