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.
November 2004, A.M. Cohen constructed this CD from parts of the former
experimental CD linalg5 by adding the symbol sparse. The other part of the
former linalg5 have gone to linalg5s. The descriptions have been slightly
edited.
This symbol denotes a unary function which is used to construct the
(nxn) identity matrix where n is the single argument. The argument n must be a
natural number.
Commented Mathematical property (CMP):
for all M | identity(rowcount M) * M = M * identity(columncount M) = M
This symbol denotes a function with two arguments, m and n, which should be natural
numbers. When applied to m and n, it represents the m x n zero matrix.
Commented Mathematical property (CMP):
for all M | M + zero(rowcount M,columncount M) = M
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]
This symbol represents a square matrix which is a scalar constant times the
identity matrix. It should take two arguments, the first
and second specify 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, is equal to
s * identity(n).
This symbol represents a matrix which has all entries of the same
value. It takes three arguments, the first is the rowcount of the matrix,
the second is the column count, and
the third is the constant which determines every element.
Commented Mathematical property (CMP):
The rank of a non-zero constant matrix is equal to 1.
This symbol represents a square (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.
The sparse symbol is a constructor for sparse matrices. It is (n+1)-ary,
where the first argument is the rowcount (row dimension) of the matrix, the
second
argument is the columncount (column dimension) of the matrix and
every following argument specifies a possibly non-zero element in the
following way. The argument is a list which should have length three.
The first element in the list is the row index, the second element is the
column index (one based), whilst the third
element in the list is the value. Every other element of the matrix is
implicitly zero.
Example:
This example shows a representation of the sparse matrix
[ [0,0,0],
[1,0,7]
]
