OpenMath

OpenMath is an extensible standard for representing the semantics of mathematical objects.

https://openmath.org

OpenMath Content Dictionary: linalg5

Canonical URL:
http://www.openmath.org/cd/linalg5.ocd
CD Base:
http://www.openmath.org/cd
CD File:
linalg5.ocd
CD as XML Encoded OpenMath:
linalg5.omcd
Defines:
Hermitian, anti-Hermitian, banded, constant, diagonal_matrix, identity, lower-Hessenberg, lower-triangular, scalar, skew-symmetric, symmetric, tridiagonal, upper-Hessenberg, upper-triangular, zero
Date:
2004-05-11
Version:
3 (Revision 2)
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.

     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.

  Author: OpenMath Consortium
  SourceURL: https://github.com/OpenMath/CDs

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

identity

Role:
application
Description:

This symbol denotes a unary function which is used to construct an (nxn) identity matrix where n is the single positive integral argument.

Commented Mathematical property (CMP):
for all M | identity(rowcount M) * M = M * identity(columncount M) = M
Formal Mathematical property (FMP):
∀   M . identity ⁡ ( rowcount ⁡ ( M ) ) ⁢ M = M ∧ M ⁢ identity ⁡ ( columncount ⁡ ( M ) ) = M
Example:
A representation of the 2x2 identity matrix [[1,0],[0,1]]
1 0 0 1
Signatures:
sts

[Next: zero] [Last: tridiagonal] [Top]

zero

Role:
application
Description:

This symbol denotes a function with two integral arguments m,n which is used to construct an (mxn) zero matrix.

Commented Mathematical property (CMP):
for all M | M + zero(rowcount M,columncount M) = M
Formal Mathematical property (FMP):
∀   M . M + zero ⁡ ( rowcount ⁡ ( M ) , columncount ⁡ ( M ) ) = M
Commented Mathematical property (CMP):
for all M | zero(rowcount M,rowcount M) * M = M * zero(columncount M,columncount M) = zero(rowcount M,columncount M)
Formal Mathematical property (FMP):
∀   M . zero ⁡ ( rowcount ⁡ ( M ) , rowcount ⁡ ( M ) ) ⁢ M = zero ⁡ ( rowcount ⁡ ( M ) , columncount ⁡ ( M ) ) ∧ M ⁢ zero ⁡ ( columncount ⁡ ( M ) , columncount ⁡ ( M ) ) = zero ⁡ ( rowcount ⁡ ( M ) , columncount ⁡ ( M ) )
Example:
A representation of the 2x2 zero matrix [[0,0],[0,0]]
0 0 0 0
Signatures:
sts

[Next: diagonal_matrix] [Previous: identity] [Top]

diagonal_matrix

Role:
application
Description:

This symbol denotes an n_ary function which is used to construct an (nxn) diagonal matrix, that is a matrix where every non-diagonal element is zero, the diagonal elements are equal to the n arguments.

Commented Mathematical property (CMP):
given a diagonal matrix, it is equal to its transpose
Example:
The diagonal matrix with diagonal elements [1,2,3]
1 0 0 0 2 0 0 0 3
Signatures:
sts

[Next: scalar] [Previous: zero] [Top]

scalar

Role:
application
Description:

This symbol represents a matrix which is a scalar constant times the identity matrix. It should take two arguments, the first specifes the number of rows and columns in the matrix respectively and the third specifies the scalar multiplier.

Commented Mathematical property (CMP):
the scalar matrix of size n, where the scalar multiple is s = s * identity(n)
Formal Mathematical property (FMP):
scalar ⁡ ( n , s ) = s ⁢ identity ⁡ ( n )
Example:
1.5 0 0 0 0 1.5 0 0 0 0 1.5 0 0 0 0 1.5
Signatures:
sts

[Next: constant] [Previous: diagonal_matrix] [Top]

constant

Role:
application
Description:

This symbol represents a matrix which has all entries of the same value. It takes two arguments, the first is the size of the matrix, the second is the constant which determines every element.

Commented Mathematical property (CMP):
the rank of a non-zero constant matrix = 1
Formal Mathematical property (FMP):
v ≠ 0 ⇒ rank ⁡ ( constant ⁡ ( n , v ) ) = 1
Signatures:
sts

[Next: banded] [Previous: scalar] [Top]

banded

Role:
application
Description:

This symbol represents a (p,q) banded matrix, it takes one argument. A (p,q) banded matrix should always be square. The lower non-zero subdiagonal is the first element of the argument, whilst the highest non-zero super-diagonal is given by the last element of the argument. The argument determines the band of possibly non-zero entries which are positioned around the diagonal. It should be a vector of vectors, we note that they will not all be the same length, however the length of the vectors determine p and q. The longest element specifies the diagonal of the matrix and hence the size of the matrix. Every element not in the band is zero.

Example:
A specification of the (2,1) banded matrix: [ [1 2 3 0 0] [4 5 6 7 0] [0 8 9 10 11] [0 0 12 13 14] [0 0 0 15 16]]
1 2 3 0 0 4 5 6 7 0 0 8 9 10 11 0 0 12 13 14 0 0 0 15 16
Signatures:
sts

[Next: symmetric] [Previous: constant] [Top]

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):
symmetric ⁡ ( VV 1 ) + symmetric ⁡ ( VV 1 ) T = symmetric ⁡ ( VV 2 )
Commented Mathematical property (CMP):
for a symmetric matrix M, M = transpose M
Formal Mathematical property (FMP):
symmetric ⁡ ( VV ) = symmetric ⁡ ( VV ) T
Commented Mathematical property (CMP):
the dimension of a symmetric matrix = the length of the vector which defines it
Formal Mathematical property (FMP):
rowcount ⁡ ( symmetric ⁡ ( VV ) ) = size ⁡ ( VV )
Example:
An example to represent the symmetric matrix: [[1,2,3,4] [2,5,6,7] [3,6,8,9] [4,7,9,10]]
1 2 3 4 2 5 6 7 3 6 8 9 4 7 9 10
Signatures:
sts

[Next: skew-symmetric] [Previous: banded] [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):
skew-symmetric ⁡ ( VV ) i i = 0
Commented Mathematical property (CMP):
for a skew-symmetric matrix M, M = - transpose M
Formal Mathematical property (FMP):
skew-symmetric ⁡ ( VV ) = - skew-symmetric ⁡ ( VV ) 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]]
0 2 3 4 - 2 0 6 7 - 3 - 6 0 9 - 4 - 7 - 9 0
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):
Hermitian ⁡ ( VV ) i i ∈ R
Example:
An example to describe the Hermitian matrix: [[1 , 2+2i] [2-2i, 3 ]]
1 2 + 2 ⁢ i 2 + 2 ⁢ i ¯ 3
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. This rules implies 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):
anti-Hermitian ⁡ ( VV ) i i = 0
Example:
An example to describe the anti-Hermitian matrix: [[0 , 1+i] [-1+i , 0 ]]
0 1 + i - 1 + i ¯ 0
Signatures:
sts

[Next: upper-triangular] [Previous: Hermitian] [Top]

upper-triangular

Role:
application
Description:

This symbol represents an upper-triangular matrix, it takes one argument. The argument should be a vector of vectors of elements of the matrix.

Commented Mathematical property (CMP):
the product of two upper-triangular matrices is upper-triangular
Formal Mathematical property (FMP):
upper-triangular ⁡ ( VV 1 ) = upper-triangular ⁡ ( VV 2 ) ⁢ upper-triangular ⁡ ( VV 3 )
Example:
An example to describe the upper triangular matrix: [[1,2,3] [0,4,5] [0,0,6]]
1 2 3 0 4 5 0 0 6
Signatures:
sts

[Next: lower-triangular] [Previous: anti-Hermitian] [Top]

lower-triangular

Role:
application
Description:

This symbol represents a lower-triangular matrix, it takes one argument. The argument should be a vector of vectors of elements of the matrix.

Commented Mathematical property (CMP):
the product of two lower-triangular matrices is lower-triangular
Formal Mathematical property (FMP):
lower-triangular ⁡ ( VV 1 ) = lower-triangular ⁡ ( VV 2 ) ⁢ lower-triangular ⁡ ( VV 3 )
Example:
An example to describe the lower triangular matrix: [[1,0,0] [2,3,0] [4,5,6]]
1 0 0 2 3 0 4 5 6
Signatures:
sts

[Next: upper-Hessenberg] [Previous: upper-triangular] [Top]

upper-Hessenberg

Role:
application
Description:

This symbol represents an upper-Hessenberg matrix, it takes one argument, the argument is a vector of vectors representing the non-zero elements. The first element of the argument specifies the value of the first subdiagonal, the subsequent elements specify the value of the diagonal and subsequent super-diagonals, all other elements are zero.

Example:
A specification of an upper-Hessenberg matrix of dimension 5: [[1 2 3 0 0] [4 5 6 7 0] [0 8 9 10 11] [0 0 12 13 14] [0 0 0 15 16]]
1 2 3 0 0 4 5 6 7 0 0 8 9 10 11 0 0 12 13 14 0 0 0 15 16
Commented Mathematical property (CMP):
the transpose of an upper-Hessenberg matrix is lower-Hessenberg
Formal Mathematical property (FMP):
upper-Hessenberg ⁡ ( VV 1 ) T = lower-Hessenberg ⁡ ( VV 2 )
Signatures:
sts

[Next: lower-Hessenberg] [Previous: lower-triangular] [Top]

lower-Hessenberg

Role:
application
Description:

This symbol represents a lower-Hessenberg matrix, it takes one argument, the argument is a vector of vectors representing the non-zero elements. The first element of the argument specifies the value of the first super-diagonal, the subsequent elements specify the value of the diagonal and subsequent subdiagonals, all other elements are zero.

Example:
A specification of a lower-Hessenberg matrix of dimension 5: [[1 2 0 0 0] [3 4 5 0 0] [6 7 8 9 0] [0 10 11 12 13] [0 0 14 15 16]]
1 2 0 0 0 3 4 5 0 0 6 7 8 9 0 0 10 11 12 13 0 0 14 15 16
Commented Mathematical property (CMP):
the transpose of a lower-Hessenberg matrix is upper-Hessenberg
Formal Mathematical property (FMP):
lower-Hessenberg ⁡ ( VV 1 ) T = upper-Hessenberg ⁡ ( VV 2 )
Signatures:
sts

[Next: tridiagonal] [Previous: upper-Hessenberg] [Top]

tridiagonal

Role:
application
Description:

This symbol represents a tridiagonal matrix, it takes one argument which should be a vector of vectors which should have three elements. These should be vectors representing the sub-diagonal, the diagonal and the super-diagonal in that order.

Commented Mathematical property (CMP):
a tridiagonal matrix is a (1,1) banded matrix
Formal Mathematical property (FMP):
size ⁡ ( VV ) = 3 ∧ size ⁡ ( VV 2 ) = size ⁡ ( VV 1 ) + 1 ∧ tridiagonal ⁡ ( VV ) = banded ⁡ ( VV )
Commented Mathematical property (CMP):
The product of two tridiagonal matrices is tridiagonal
Formal Mathematical property (FMP):
tridiagonal ⁡ ( VV 1 ) ⁢ tridiagonal ⁡ ( VV 2 ) = tridiagonal ⁡ ( VV 3 )
Signatures:
sts

[First: identity] [Previous: lower-Hessenberg] [Top]