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:

anti_Hermitian, Hermitian, skew_symmetric, symmetric

Date:

2004-05-11

Version:

3 (Revision 1)

Review Date:

2006-03-30

Status:

experimental

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          work to the original source.
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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):

xml prefix mathml

<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>

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

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

Commented Mathematical property (CMP):

for a symmetric matrix M, M = transpose M

Formal Mathematical property (FMP):

xml prefix mathml

<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>

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

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

Commented Mathematical property (CMP):

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

Formal Mathematical property (FMP):

xml prefix mathml

<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>

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

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

Example:

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

xml prefix mathml

<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>

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

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

Signatures:

sts

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):

xml prefix mathml

<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>

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

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

Commented Mathematical property (CMP):

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

Formal Mathematical property (FMP):

xml prefix mathml

<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>

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

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

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]]

xml prefix mathml

<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>

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

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

Signatures:

sts

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):

xml prefix mathml

<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>

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

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

Example:

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

xml prefix mathml

<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>

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

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

Signatures:

sts

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):

xml prefix mathml

<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>

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

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

Example:

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

xml prefix mathml

<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>

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

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

Signatures:

sts