OpenMath Content Dictionary: linalgsym1

Canonical URL:

http://www.openmath.org/cd/linalgsym1.ocd

CD Base:

http://www.openmath.org/cd

CD File:

linalgsym1.ocd

CD as XML Encoded OpenMath:

linalgsym1.omcd

Defines:

Hermitian, anti_Hermitian, skew_symmetric, symmetric

Date:

2004-05-11

Version:

3 (Revision 1)

Review Date:

2006-03-30

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.

     The copyright holder grants you permission to redistribute this 
     document freely as a verbatim copy. Furthermore, the copyright
     holder permits you to develop any derived work from this document
     provided that the following conditions are met.
       a) The derived work acknowledges the fact that it is derived from
          this document, and maintains a prominent reference in the 
          work to the original source.
       b) The fact that the derived work is not the original OpenMath 
          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
          the OpenMath Society.  In particular, if there is a Content 
          Dictionary Group whose name is, for example, `math' containing
          Content Dictionaries named `math1', `math2' etc., then you should 
          not name a derived Content Dictionary `mathN' where N is an integer.
          However you are free to name it `private_mathN' or some such.  This
          is because the names `mathN' may be used by the OpenMath Society
          for future extensions.
       c) The derived work is distributed under terms that allow the
          compilation of derived works, but keep paragraphs a) and b)
          intact.  The simplest way to do this is to distribute the derived
          work under the OpenMath license, but this is not a requirement.
     If you have questions about this license please contact the OpenMath
     society at http://www.openmath.org.

This CD contains symbols which represent a number of special types of matrix related to symmetry.

November 2004, A.M. Cohen removed the symmetry alien parts from this CD and renamed it from linalg5 to linalgsym1.

---

symmetric

Role:

application

Description:

This symbol represents a symmetric matrix, it takes one argument. The argument should be a vector of vectors of elements of the matrix. For j>=i the ij'th element of the matrix is the (j-i+1)'th element of the i'th element of the argument. This determines the upper triangle of the matrix, the lower triangle is specified by the rule M = transpose M.

Commented Mathematical property (CMP):

the sum of a symmetric matrix and its transpose is symmetric

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="plus"/>
      <OMA>
        <OMS cd="linalgsym1" name="symmetric"/>
	<OMV name="VV1"/>
      </OMA>
      <OMA>
        <OMS cd="linalg1" name="transpose"/>
	<OMA>
	  <OMS cd="linalgsym1" name="symmetric"/>
	  <OMV name="VV1"/>
	</OMA>
      </OMA>
    </OMA>
    <OMA>
      <OMS cd="linalgsym1" name="symmetric"/>
      <OMV name="VV2"/>
    </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">plus</csymbol>
   <apply><csymbol cd="linalgsym1">symmetric</csymbol><ci>VV1</ci></apply>
   <apply><csymbol cd="linalg1">transpose</csymbol>
    <apply><csymbol cd="linalgsym1">symmetric</csymbol><ci>VV1</ci></apply>
   </apply>
  </apply>
  <apply><csymbol cd="linalgsym1">symmetric</csymbol><ci>VV2</ci></apply>
 </apply>
</math>

Prefix

eq (plus (symmetric ( VV1) , transpose (symmetric ( VV1) ) ) , symmetric ( VV2) )

Popcorn

linalgsym1.symmetric($VV1) + linalg1.transpose(linalgsym1.symmetric($VV1)) = linalgsym1.symmetric($VV2)

Rendered Presentation MathML

$$\mathrm{symmetric}\left({\mathrm{VV}}_1\right)+{\mathrm{symmetric}\left({\mathrm{VV}}_1\right)}^T=\mathrm{symmetric}\left({\mathrm{VV}}_2\right)$$

Commented Mathematical property (CMP):

for a symmetric matrix M, M = transpose M

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="linalgsym1" name="symmetric"/>
      <OMV name="VV"/>
    </OMA>
    <OMA>
      <OMS cd="linalg1" name="transpose"/>
      <OMA>
        <OMS cd="linalgsym1" name="symmetric"/>
        <OMV name="VV"/>
      </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="linalgsym1">symmetric</csymbol><ci>VV</ci></apply>
  <apply><csymbol cd="linalg1">transpose</csymbol>
   <apply><csymbol cd="linalgsym1">symmetric</csymbol><ci>VV</ci></apply>
  </apply>
 </apply>
</math>

Prefix

eq (symmetric ( VV) , transpose (symmetric ( VV) ) )

Popcorn

linalgsym1.symmetric($VV) = linalg1.transpose(linalgsym1.symmetric($VV))

Rendered Presentation MathML

$$\mathrm{symmetric}\left(\mathrm{VV}\right)={\mathrm{symmetric}\left(\mathrm{VV}\right)}^T$$

Commented Mathematical property (CMP):

the dimension of a symmetric matrix = the length of the vector which defines it

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="linalgsym1" name="symmetric"/>
	<OMV name="VV"/>
      </OMA>
    </OMA>
    <OMA>
      <OMS cd="linalg3" name="size"/>
      <OMV name="VV"/>
    </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="linalgsym1">symmetric</csymbol><ci>VV</ci></apply>
  </apply>
  <apply><csymbol cd="linalg3">size</csymbol><ci>VV</ci></apply>
 </apply>
</math>

Prefix

eq (rowcount (symmetric ( VV) ) , size ( VV) )

Popcorn

linalg3.rowcount(linalgsym1.symmetric($VV)) = linalg3.size($VV)

Rendered Presentation MathML

$$\mathrm{rowcount}\left(\mathrm{symmetric}\left(\mathrm{VV}\right)\right)=\mathrm{size}\left(\mathrm{VV}\right)$$

Example:

An example to represent the symmetric matrix: [[1,2,3,4] [2,5,6,7] [3,6,8,9] [4,7,9,10]]

OpenMath XML (source)

<OMOBJ xmlns="http://www.openmath.org/OpenMath" version="2.0" cdbase="http://www.openmath.org/cd">
  <OMA>
    <OMS cd="linalgsym1" name="symmetric"/>
    <OMA>
      <OMS cd="linalg2" name="vector"/>
      <OMA>
        <OMS cd="linalg2" name="vector"/>
	<OMI> 1 </OMI>	
	<OMI> 2 </OMI>	
	<OMI> 3 </OMI>	
	<OMI> 4 </OMI>	
      </OMA>
      <OMA>
        <OMS cd="linalg2" name="vector"/>
	<OMI> 5 </OMI>	
	<OMI> 6 </OMI>	
	<OMI> 7 </OMI>	
      </OMA>
      <OMA>
        <OMS cd="linalg2" name="vector"/>
	<OMI> 8 </OMI>	
	<OMI> 9 </OMI>	
      </OMA>
      <OMA>
        <OMS cd="linalg2" name="vector"/>
	<OMI> 10 </OMI>
      </OMA>
    </OMA>
  </OMA>
</OMOBJ>

Strict Content MathML

<math xmlns="http://www.w3.org/1998/Math/MathML">
 <apply><csymbol cd="linalgsym1">symmetric</csymbol>
  <apply><csymbol cd="linalg2">vector</csymbol>
   <apply><csymbol cd="linalg2">vector</csymbol>
    <cn type="integer">1</cn>
    <cn type="integer">2</cn>
    <cn type="integer">3</cn>
    <cn type="integer">4</cn>
   </apply>
   <apply><csymbol cd="linalg2">vector</csymbol>
    <cn type="integer">5</cn>
    <cn type="integer">6</cn>
    <cn type="integer">7</cn>
   </apply>
   <apply><csymbol cd="linalg2">vector</csymbol>
    <cn type="integer">8</cn>
    <cn type="integer">9</cn>
   </apply>
   <apply><csymbol cd="linalg2">vector</csymbol><cn type="integer">10</cn></apply>
  </apply>
 </apply>
</math>

Prefix

symmetric (vector (vector ( 1 , 2 , 3 , 4 ) , vector ( 5 , 6 , 7 ) , vector ( 8 , 9 ) , vector ( 10 ) ) )

Popcorn

linalgsym1.symmetric(linalg2.vector(linalg2.vector(1, 2, 3, 4), linalg2.vector(5, 6, 7), linalg2.vector(8, 9), linalg2.vector(10)))

Rendered Presentation MathML

$$\mathrm{symmetric}\left(((1,2,3,4),(5,6,7),(8,9),\left(10\right))\right)$$

Signatures:

sts

---

[Next: skew_symmetric] [Last: anti_Hermitian] [Top]

---

skew_symmetric

Role:

application

Description:

This symbol represents a skew-symmetric matrix, it takes one argument. The argument should be a vector of vectors of elements of the matrix. For j>i the ij'th element of the matrix is the (j-i+1)'th element of the i'th element of the argument. This determines the elements above the diagonal of the matrix, the elements below the diagonal of the matrix must conform to the rule M = - transpose M. This rule implies that the elements on the diagonal must be equal to 0, therefore we do not include these in the argument.

Commented Mathematical property (CMP):

The elements on the diagonal of a skew-symmetric matrix are zero

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="linalg1" name="matrix_selector"/>
      <OMV name="i"/>
      <OMV name="i"/>
      <OMA>
        <OMS cd="linalgsym1" name="skew_symmetric"/>
	<OMV name="VV"/>
      </OMA>
    </OMA>
    <OMS cd="alg1" name="zero"/>
  </OMA>
</OMOBJ>

Strict Content MathML

<math xmlns="http://www.w3.org/1998/Math/MathML">
 <apply><csymbol cd="relation1">eq</csymbol>
  <apply><csymbol cd="linalg1">matrix_selector</csymbol>
   <ci>i</ci>
   <ci>i</ci>
   <apply><csymbol cd="linalgsym1">skew_symmetric</csymbol><ci>VV</ci></apply>
  </apply>
  <csymbol cd="alg1">zero</csymbol>
 </apply>
</math>

Prefix

eq (matrix_selector ( i, i, skew_symmetric ( VV) ) , zero)

Popcorn

linalg1.matrix_selector($i, $i, linalgsym1.skew_symmetric($VV)) = alg1.zero

Rendered Presentation MathML

$${\mathrm{skew\_symmetric}\left(\mathrm{VV}\right)}_{\left(i,i\right)}=0$$

Commented Mathematical property (CMP):

for a skew-symmetric matrix M, M = - transpose M

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="linalgsym1" name="skew_symmetric"/>
      <OMV name="VV"/>
    </OMA>
    <OMA>
      <OMS cd="arith1" name="unary_minus"/>
      <OMA>
        <OMS cd="linalg1" name="transpose"/>
        <OMA>
          <OMS cd="linalgsym1" name="skew_symmetric"/>
          <OMV name="VV"/>
        </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="linalgsym1">skew_symmetric</csymbol><ci>VV</ci></apply>
  <apply><csymbol cd="arith1">unary_minus</csymbol>
   <apply><csymbol cd="linalg1">transpose</csymbol>
    <apply><csymbol cd="linalgsym1">skew_symmetric</csymbol><ci>VV</ci></apply>
   </apply>
  </apply>
 </apply>
</math>

Prefix

eq (skew_symmetric ( VV) , unary_minus (transpose (skew_symmetric ( VV) ) ) )

Popcorn

linalgsym1.skew_symmetric($VV) = -(linalg1.transpose(linalgsym1.skew_symmetric($VV)))

Rendered Presentation MathML

$$\mathrm{skew\_symmetric}\left(\mathrm{VV}\right)=-{\mathrm{skew\_symmetric}\left(\mathrm{VV}\right)}^T$$

Example:

An example to represent the skew-symmetric matrix: [[ 0, 2, 3, 4] [-2, 0, 6, 7] [-3,-6, 0, 9] [-4,-7,-9, 0]]

OpenMath XML (source)

<OMOBJ xmlns="http://www.openmath.org/OpenMath" version="2.0" cdbase="http://www.openmath.org/cd">
  <OMA>
    <OMS cd="linalgsym1" name="skew_symmetric"/>
    <OMA>
      <OMS cd="linalg2" name="vector"/>
      <OMA>
        <OMS cd="linalg2" name="vector"/>
	<OMI> 2 </OMI>	
	<OMI> 3 </OMI>	
	<OMI> 4 </OMI>	
      </OMA>
      <OMA>
        <OMS cd="linalg2" name="vector"/>
	<OMI> 6 </OMI>	
	<OMI> 7 </OMI>	
      </OMA>
      <OMA>
        <OMS cd="linalg2" name="vector"/>
	<OMI> 9 </OMI>	
      </OMA>
    </OMA>
  </OMA>
</OMOBJ>

Strict Content MathML

<math xmlns="http://www.w3.org/1998/Math/MathML">
 <apply><csymbol cd="linalgsym1">skew_symmetric</csymbol>
  <apply><csymbol cd="linalg2">vector</csymbol>
   <apply><csymbol cd="linalg2">vector</csymbol>
    <cn type="integer">2</cn>
    <cn type="integer">3</cn>
    <cn type="integer">4</cn>
   </apply>
   <apply><csymbol cd="linalg2">vector</csymbol>
    <cn type="integer">6</cn>
    <cn type="integer">7</cn>
   </apply>
   <apply><csymbol cd="linalg2">vector</csymbol><cn type="integer">9</cn></apply>
  </apply>
 </apply>
</math>

Prefix

skew_symmetric (vector (vector ( 2 , 3 , 4 ) , vector ( 6 , 7 ) , vector ( 9 ) ) )

Popcorn

linalgsym1.skew_symmetric(linalg2.vector(linalg2.vector(2, 3, 4), linalg2.vector(6, 7), linalg2.vector(9)))

Rendered Presentation MathML

$$\mathrm{skew\_symmetric}\left(((2,3,4),(6,7),\left(9\right))\right)$$

Signatures:

sts

---

[Next: Hermitian] [Previous: symmetric] [Top]

---

Hermitian

Role:

application

Description:

This symbol represents a Hermitian matrix, it takes one argument. The argument should be a vector of vectors of values which determine the upper triangle of the matrix. The lower triangle of the matrix is specified by the following relation: M^* = transpose(M), were M^* denotes the matrix consisting of all the complex conjugates of M.

Commented Mathematical property (CMP):

The complex conjugate of a Hermitian matrix equals its transpose

Commented Mathematical property (CMP):

The diagonal elements of a Hermitian matrix will be real

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="set1" name="in"/>
    <OMA>
      <OMS cd="linalg1" name="matrix_selector"/>
      <OMV name="i"/>
      <OMV name="i"/>
      <OMA>
        <OMS cd="linalgsym1" name="Hermitian"/>
	<OMV name="VV"/>
      </OMA>
    </OMA>
    <OMS cd="setname1" name="R"/>
  </OMA>
</OMOBJ>

Strict Content MathML

<math xmlns="http://www.w3.org/1998/Math/MathML">
 <apply><csymbol cd="set1">in</csymbol>
  <apply><csymbol cd="linalg1">matrix_selector</csymbol>
   <ci>i</ci>
   <ci>i</ci>
   <apply><csymbol cd="linalgsym1">Hermitian</csymbol><ci>VV</ci></apply>
  </apply>
  <csymbol cd="setname1">R</csymbol>
 </apply>
</math>

Prefix

in (matrix_selector ( i, i, Hermitian ( VV) ) , R)

Popcorn

set1.in(linalg1.matrix_selector($i, $i, linalgsym1.Hermitian($VV)), setname1.R)

Rendered Presentation MathML

$${\mathrm{Hermitian}\left(\mathrm{VV}\right)}_{\left(i,i\right)}\in \mathbb{R}$$

Example:

An example to describe the Hermitian matrix: [[1 , 2+2i] [2-2i, 3]]

OpenMath XML (source)

<OMOBJ xmlns="http://www.openmath.org/OpenMath" version="2.0" cdbase="http://www.openmath.org/cd">
  <OMA>
    <OMS cd="linalgsym1" name="Hermitian"/>
    <OMA>
      <OMS cd="linalg2" name="vector"/>
      <OMA>
        <OMS cd="linalg2" name="vector"/>
        <OMI>1</OMI>
	<OMA>
	  <OMS cd="complex1" name="complex_cartesian"/>
	  <OMI> 2 </OMI><OMI> 2 </OMI>
	</OMA>
      </OMA>
      <OMA>
        <OMS cd="linalg2" name="vector"/>
        <OMI>3</OMI>
      </OMA>
    </OMA>
  </OMA>
</OMOBJ>

Strict Content MathML

<math xmlns="http://www.w3.org/1998/Math/MathML">
 <apply><csymbol cd="linalgsym1">Hermitian</csymbol>
  <apply><csymbol cd="linalg2">vector</csymbol>
   <apply><csymbol cd="linalg2">vector</csymbol>
    <cn type="integer">1</cn>
    <apply><csymbol cd="complex1">complex_cartesian</csymbol>
     <cn type="integer">2</cn>
     <cn type="integer">2</cn>
    </apply>
   </apply>
   <apply><csymbol cd="linalg2">vector</csymbol><cn type="integer">3</cn></apply>
  </apply>
 </apply>
</math>

Prefix

Hermitian (vector (vector (1, complex_cartesian ( 2 , 2 ) ) , vector (3) ) )

Popcorn

linalgsym1.Hermitian(linalg2.vector(linalg2.vector(1, 2 | 2), linalg2.vector(3)))

Rendered Presentation MathML

$$\mathrm{Hermitian}\left(((1,2+2i),\left(3\right))\right)$$

Signatures:

sts

---

[Next: anti_Hermitian] [Previous: skew_symmetric] [Top]

---

anti_Hermitian

Role:

application

Description:

This symbol represents an anti-Hermitian matrix, it takes one argument. The argument should be a vector of vectors of values which determine the upper triangle of the matrix. The lower triangle of the matrix is specified by the following relation: - M^* = transpose(M), were M^* denotes the matrix consisting of all the complex conjugates of M. These rules imply that the main diagonal is zero, therefore the argument should not include it.

Commented Mathematical property (CMP):

The complex conjugate of an anti-Hermitian matrix equals minus its transpose

Commented Mathematical property (CMP):

an anti-hermitian matrix will have zero on the diagonal

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="linalg1" name="matrix_selector"/>
      <OMV name="i"/>
      <OMV name="i"/>
      <OMA>
        <OMS cd="linalgsym1" name="anti_Hermitian"/>
	<OMV name="VV"/>
      </OMA>
    </OMA>
    <OMS cd="alg1" name="zero"/>
  </OMA>
</OMOBJ>

Strict Content MathML

<math xmlns="http://www.w3.org/1998/Math/MathML">
 <apply><csymbol cd="relation1">eq</csymbol>
  <apply><csymbol cd="linalg1">matrix_selector</csymbol>
   <ci>i</ci>
   <ci>i</ci>
   <apply><csymbol cd="linalgsym1">anti_Hermitian</csymbol><ci>VV</ci></apply>
  </apply>
  <csymbol cd="alg1">zero</csymbol>
 </apply>
</math>

Prefix

eq (matrix_selector ( i, i, anti_Hermitian ( VV) ) , zero)

Popcorn

linalg1.matrix_selector($i, $i, linalgsym1.anti_Hermitian($VV)) = alg1.zero

Rendered Presentation MathML

$${\mathrm{anti\_Hermitian}\left(\mathrm{VV}\right)}_{\left(i,i\right)}=0$$

Example:

An example to describe the anti-Hermitian matrix: [[0 , 1+i] [-1+i , 0 ]]

OpenMath XML (source)

<OMOBJ xmlns="http://www.openmath.org/OpenMath" version="2.0" cdbase="http://www.openmath.org/cd">
  <OMA>
    <OMS cd="linalgsym1" name="anti_Hermitian"/>
    <OMA>
      <OMS cd="linalg2" name="vector"/>
    <OMA>
      <OMS cd="linalg2" name="vector"/>
      <OMA>
        <OMS cd="complex1" name="complex_cartesian"/>
	<OMI> 1 </OMI><OMI> 1 </OMI>
      </OMA>
    </OMA>
  </OMA>
  </OMA>
</OMOBJ>

Strict Content MathML

<math xmlns="http://www.w3.org/1998/Math/MathML">
 <apply><csymbol cd="linalgsym1">anti_Hermitian</csymbol>
  <apply><csymbol cd="linalg2">vector</csymbol>
   <apply><csymbol cd="linalg2">vector</csymbol>
    <apply><csymbol cd="complex1">complex_cartesian</csymbol>
     <cn type="integer">1</cn>
     <cn type="integer">1</cn>
    </apply>
   </apply>
  </apply>
 </apply>
</math>

Prefix

anti_Hermitian (vector (vector (complex_cartesian ( 1 , 1 ) ) ) )

Popcorn

linalgsym1.anti_Hermitian(linalg2.vector(linalg2.vector(1 | 1)))

Rendered Presentation MathML

$$\mathrm{anti\_Hermitian}\left(\left(\left(1+i\right)\right)\right)$$

Signatures:

sts

---

[First: symmetric] [Previous: Hermitian] [Top]