OpenMath Content Dictionary: linalg4

Canonical URL:

http://www.openmath.org/cd/linalg4.ocd

CD Base:

http://www.openmath.org/cd

CD File:

linalg4.ocd

CD as XML Encoded OpenMath:

linalg4.omcd

Defines:

characteristic_eqn, columncount, eigenvalue, eigenvector, rank, rowcount, size

Date:

2004-03-30

Version:

3 (Revision 1)

Review Date:

2017-12-31

Status:

experimental

---

     This document is distributed in the hope that it will be useful, 
     but WITHOUT ANY WARRANTY; without even the implied warranty of 
     MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE.

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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 
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          document is stated prominently in the derived work.  Moreover if
          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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          Dictionary Group whose name is, for example, `math' containing
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  Author: OpenMath Consortium
  SourceURL: https://github.com/OpenMath/CDs
            

This CD defines symbols for basic linear algebra.

Regardless of the way of forming vectors and matrices, this CD deals with eigenvalues, eigenvectors and related concepts.

---

eigenvalue

Role:

application

Description:

This symbol represents the eigenvalue of a matrix. It takes two arguments the first should be the matrix, the second should be an index to specify the eigenvalue. The ordering imposed on the eigenvalues is first on the modulus of the value, and second on the argument of the value. A definition of eigenvalue is given in Elementary Linear Algebra, Stanley I. Grossman in Definition 1 of chapter 6, page 533.

Commented Mathematical property (CMP):

A*eigenvector(A,i) = eigenvalue(A,i)*eigenvector(A,i)

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="linalg4" name="eigenvector"/>
	<OMV name="A"/>
	<OMV name="i"/>
      </OMA>
    </OMA>
    <OMA>
      <OMS cd="arith1" name="times"/>
      <OMA>
        <OMS cd="linalg4" name="eigenvalue"/>
	<OMV name="A"/>
	<OMV name="i"/>
      </OMA>
      <OMA>
        <OMS cd="linalg4" name="eigenvector"/>
	<OMV name="A"/>
	<OMV name="i"/>
      </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="linalg4">eigenvector</csymbol><ci>A</ci><ci>i</ci></apply>
  </apply>
  <apply><csymbol cd="arith1">times</csymbol>
   <apply><csymbol cd="linalg4">eigenvalue</csymbol><ci>A</ci><ci>i</ci></apply>
   <apply><csymbol cd="linalg4">eigenvector</csymbol><ci>A</ci><ci>i</ci></apply>
  </apply>
 </apply>
</math>

Prefix

eq (times ( A, eigenvector ( A, i) ) , times (eigenvalue ( A, i) , eigenvector ( A, i) ) )

Popcorn

$A * linalg4.eigenvector($A, $i) = linalg4.eigenvalue($A, $i) * linalg4.eigenvector($A, $i)

Rendered Presentation MathML

$$A\mathrm{eigenvector}(A,i)=\mathrm{eigenvalue}(A,i)\mathrm{eigenvector}(A,i)$$

Signatures:

sts

---

[Next: eigenvector] [Last: columncount] [Top]

---

eigenvector

Role:

application

Description:

This symbol represents the eigenvector of a matrix. It takes two arguments the first should be the matrix, the second should be an index to specify which eigenvalue this eigenvector should be paired with. The ordering is as given in the eigenvalue symbol. A definition of eigenvector is given in Elementary Linear Algebra, Stanley I. Grossman in Definition 1 of chapter 6, page 533.

Commented Mathematical property (CMP):

A*eigenvector(A) = eigenvalue(A)*eigenvector(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="arith1" name="times"/>
      <OMV name="A"/>
      <OMA>
        <OMS cd="linalg4" name="eigenvector"/>
	<OMV name="A"/>
	<OMV name="i"/>
      </OMA>
    </OMA>
    <OMA>
      <OMS cd="arith1" name="times"/>
      <OMA>
        <OMS cd="linalg4" name="eigenvalue"/>
	<OMV name="A"/>
	<OMV name="i"/>
      </OMA>
      <OMA>
        <OMS cd="linalg4" name="eigenvector"/>
	<OMV name="A"/>
	<OMV name="i"/>
      </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="linalg4">eigenvector</csymbol><ci>A</ci><ci>i</ci></apply>
  </apply>
  <apply><csymbol cd="arith1">times</csymbol>
   <apply><csymbol cd="linalg4">eigenvalue</csymbol><ci>A</ci><ci>i</ci></apply>
   <apply><csymbol cd="linalg4">eigenvector</csymbol><ci>A</ci><ci>i</ci></apply>
  </apply>
 </apply>
</math>

Prefix

eq (times ( A, eigenvector ( A, i) ) , times (eigenvalue ( A, i) , eigenvector ( A, i) ) )

Popcorn

$A * linalg4.eigenvector($A, $i) = linalg4.eigenvalue($A, $i) * linalg4.eigenvector($A, $i)

Rendered Presentation MathML

$$A\mathrm{eigenvector}(A,i)=\mathrm{eigenvalue}(A,i)\mathrm{eigenvector}(A,i)$$

Signatures:

sts

---

[Next: characteristic_eqn] [Previous: eigenvalue] [Top]

---

characteristic_eqn

Role:

application

Description:

This symbol represents the polynomial which appears in the left hand side of the characteristic equation of a matrix. It takes one argument which should be the matrix. A definition of the characteristic equation is given in Elementary Linear Algebra, Stanley I. Grossman in Definition 2 of chapter 6, page 535.

Commented Mathematical property (CMP):

p(eigenvalue(A,i)) = det(A-eigenvalue(A,i)I) = 0 where p is the characteristic equation of 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="logic1" name="and"/>
    <OMA>
      <OMS cd="relation1" name="eq"/>
      <OMA>
        <OMA>
	  <OMS cd="linalg4" name="characteristic_eqn"/>
	  <OMV name="A"/>
	</OMA>
	<OMA>
	  <OMS cd="linalg4" name="eigenvalue"/>
	  <OMV name="A"/>
	  <OMV name="i"/>
	</OMA>
      </OMA>
      <OMS cd="alg1" name="zero"/>
    </OMA>
    <OMA>
      <OMS cd="relation1" name="eq"/>
      <OMA>
        <OMS cd="linalg1" name="determinant"/>
	<OMA>
	  <OMS cd="arith1" name="minus"/>
	  <OMV name="A"/>
	  <OMA>
	    <OMS cd="arith1" name="times"/>
	    <OMA>
	      <OMS cd="linalg4" name="eigenvalue"/>
	      <OMV name="A"/>
	      <OMV name="i"/>
	    </OMA>
	    <OMA>
	      <OMS cd="linalg5" name="identity"/>
	      <OMA>
	        <OMS cd="linalg4" name="rowcount"/>
		<OMV name="A"/>
	      </OMA>
	    </OMA>
	  </OMA>
	</OMA>
      </OMA>
      <OMS cd="alg1" name="zero"/>
    </OMA>
  </OMA>
</OMOBJ>

Strict Content MathML

<math xmlns="http://www.w3.org/1998/Math/MathML">
 <apply><csymbol cd="logic1">and</csymbol>
  <apply><csymbol cd="relation1">eq</csymbol>
   <apply>
    <apply><csymbol cd="linalg4">characteristic_eqn</csymbol><ci>A</ci></apply>
    <apply><csymbol cd="linalg4">eigenvalue</csymbol><ci>A</ci><ci>i</ci></apply>
   </apply>
   <csymbol cd="alg1">zero</csymbol>
  </apply>
  <apply><csymbol cd="relation1">eq</csymbol>
   <apply><csymbol cd="linalg1">determinant</csymbol>
    <apply><csymbol cd="arith1">minus</csymbol>
     <ci>A</ci>
     <apply><csymbol cd="arith1">times</csymbol>
      <apply><csymbol cd="linalg4">eigenvalue</csymbol><ci>A</ci><ci>i</ci></apply>
      <apply><csymbol cd="linalg5">identity</csymbol>
       <apply><csymbol cd="linalg4">rowcount</csymbol><ci>A</ci></apply>
      </apply>
     </apply>
    </apply>
   </apply>
   <csymbol cd="alg1">zero</csymbol>
  </apply>
 </apply>
</math>

Prefix

and (eq (characteristic_eqn ( A) (eigenvalue ( A, i) ) , zero) , eq (determinant (minus ( A, times (eigenvalue ( A, i) , identity (rowcount ( A) ) ) ) ) , zero) )

Popcorn

linalg4.characteristic_eqn($A)(linalg4.eigenvalue($A, $i)) = alg1.zero and linalg1.determinant($A - linalg4.eigenvalue($A, $i) * linalg5.identity(linalg4.rowcount($A))) = alg1.zero

Rendered Presentation MathML

$$\left(\mathrm{characteristic\_eqn}\left(A\right)\right)\left(\mathrm{eigenvalue}(A,i)\right)=0\wedge \det \left(A-\mathrm{eigenvalue}(A,i)\mathrm{identity}\left(\mathrm{rowcount}\left(A\right)\right)\right)=0$$

Signatures:

sts

---

[Next: size] [Previous: eigenvector] [Top]

---

size

Role:

application

Description:

This symbol represents the function which takes one vector argument and returns the length of that vector.

Example:

the length of the vector [1,2,3] = 3

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="linalg4" name="size"/>
      <OMA>
        <OMS cd="linalg2" name="vector"/>
        <OMI> 1 </OMI>
        <OMI> 2 </OMI>
        <OMI> 3 </OMI>
      </OMA>
    </OMA>
    <OMI> 3 </OMI>
  </OMA>
</OMOBJ>

Strict Content MathML

<math xmlns="http://www.w3.org/1998/Math/MathML">
 <apply><csymbol cd="relation1">eq</csymbol>
  <apply><csymbol cd="linalg4">size</csymbol>
   <apply><csymbol cd="linalg2">vector</csymbol>
    <cn type="integer">1</cn>
    <cn type="integer">2</cn>
    <cn type="integer">3</cn>
   </apply>
  </apply>
  <cn type="integer">3</cn>
 </apply>
</math>

Prefix

eq (size (vector ( 1 , 2 , 3 ) ) , 3 )

Popcorn

linalg4.size(linalg2.vector(1, 2, 3)) = 3

Rendered Presentation MathML

$$\mathrm{size}\left((1,2,3)\right)=3$$

Signatures:

sts

---

[Next: rank] [Previous: characteristic_eqn] [Top]

---

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 an nxn identity matrix is 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="linalg4" name="rank"/>
      <OMA>
        <OMS cd="linalg5" 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="linalg4">rank</csymbol>
   <apply><csymbol cd="linalg5">identity</csymbol><ci>n</ci></apply>
  </apply>
  <ci>n</ci>
 </apply>
</math>

Prefix

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

Popcorn

linalg4.rank(linalg5.identity($n)) = $n

Rendered Presentation MathML

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

Signatures:

sts

---

[Next: rowcount] [Previous: size] [Top]

---

rowcount

Role:

application

Description:

This symbol represents the function which takes one matrix argument and returns the number of rows in that matrix.

Example:

Specification of the number of rows in the matrix: [[1 2] [3 4] [5 6]]

OpenMath XML (source)

<OMOBJ xmlns="http://www.openmath.org/OpenMath" version="2.0" cdbase="http://www.openmath.org/cd">
  <OMA>
    <OMS cd="linalg4" name="rowcount"/>
    <OMA>
      <OMS cd="linalg2" name="matrix"/>
      <OMA>
        <OMS cd="linalg2" name="matrixrow"/>
	<OMI>1</OMI> <OMI>2</OMI>
      </OMA>
      <OMA>
        <OMS cd="linalg2" name="matrixrow"/>
	<OMI>3</OMI> <OMI>4</OMI>
      </OMA>
      <OMA>
        <OMS cd="linalg2" name="matrixrow"/>
	<OMI>5</OMI> <OMI>6</OMI>
      </OMA>
    </OMA>
  </OMA>
</OMOBJ>

Strict Content MathML

<math xmlns="http://www.w3.org/1998/Math/MathML">
 <apply><csymbol cd="linalg4">rowcount</csymbol>
  <apply><csymbol cd="linalg2">matrix</csymbol>
   <apply><csymbol cd="linalg2">matrixrow</csymbol>
    <cn type="integer">1</cn>
    <cn type="integer">2</cn>
   </apply>
   <apply><csymbol cd="linalg2">matrixrow</csymbol>
    <cn type="integer">3</cn>
    <cn type="integer">4</cn>
   </apply>
   <apply><csymbol cd="linalg2">matrixrow</csymbol>
    <cn type="integer">5</cn>
    <cn type="integer">6</cn>
   </apply>
  </apply>
 </apply>
</math>

Prefix

rowcount (matrix (matrixrow (1, 2) , matrixrow (3, 4) , matrixrow (5, 6) ) )

Popcorn

linalg4.rowcount(linalg2.matrix(linalg2.matrixrow(1, 2), linalg2.matrixrow(3, 4), linalg2.matrixrow(5, 6)))

Rendered Presentation MathML

$$\mathrm{rowcount}\left(\left(\begin{array}{cc}1& 2\\ 3& 4\\ 5& 6\end{array}\right)\right)$$

Signatures:

sts

---

[Next: columncount] [Previous: rank] [Top]

---

columncount

Role:

application

Description:

This symbol represents the function which takes one matrix argument and returns the number of columns in that matrix.

Example:

Specification of the number of columns in the matrix: [[1 2] [3 4] [5 6]]

OpenMath XML (source)

<OMOBJ xmlns="http://www.openmath.org/OpenMath" version="2.0" cdbase="http://www.openmath.org/cd">
  <OMA>
    <OMS cd="linalg4" name="columncount"/>
    <OMA>
      <OMS cd="linalg2" name="matrix"/>
      <OMA>
        <OMS cd="linalg2" name="matrixrow"/>
	<OMI>1</OMI> <OMI>2</OMI>
      </OMA>
      <OMA>
        <OMS cd="linalg2" name="matrixrow"/>
	<OMI>3</OMI> <OMI>4</OMI>
      </OMA>
      <OMA>
        <OMS cd="linalg2" name="matrixrow"/>
	<OMI>5</OMI> <OMI>6</OMI>
      </OMA>
    </OMA>
  </OMA>
</OMOBJ>

Strict Content MathML

<math xmlns="http://www.w3.org/1998/Math/MathML">
 <apply><csymbol cd="linalg4">columncount</csymbol>
  <apply><csymbol cd="linalg2">matrix</csymbol>
   <apply><csymbol cd="linalg2">matrixrow</csymbol>
    <cn type="integer">1</cn>
    <cn type="integer">2</cn>
   </apply>
   <apply><csymbol cd="linalg2">matrixrow</csymbol>
    <cn type="integer">3</cn>
    <cn type="integer">4</cn>
   </apply>
   <apply><csymbol cd="linalg2">matrixrow</csymbol>
    <cn type="integer">5</cn>
    <cn type="integer">6</cn>
   </apply>
  </apply>
 </apply>
</math>

Prefix

columncount (matrix (matrixrow (1, 2) , matrixrow (3, 4) , matrixrow (5, 6) ) )

Popcorn

linalg4.columncount(linalg2.matrix(linalg2.matrixrow(1, 2), linalg2.matrixrow(3, 4), linalg2.matrixrow(5, 6)))

Rendered Presentation MathML

$$\mathrm{columncount}\left(\left(\begin{array}{cc}1& 2\\ 3& 4\\ 5& 6\end{array}\right)\right)$$

Signatures:

sts

---

[First: eigenvalue] [Previous: rowcount] [Top]