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The Abelian group of matrices
Dom::MatrixGroup(m, n, <R>)
Dom::MatrixGroup(m, n, R)(Array)
Dom::MatrixGroup(m, n, R)(Matrix)
Dom::MatrixGroup(m, n, R)(<m, n>)
Dom::MatrixGroup(m, n, R)(<m, n>, List)
Dom::MatrixGroup(m, n, R)(<m, n>, ListOfRows)
Dom::MatrixGroup(m, n, R)(<m, n>, f)
Dom::MatrixGroup(m, n, R)(<m, n>, List, <Diagonal>)
Dom::MatrixGroup(m, n, R)(<m, n>, g, <Diagonal>)
Dom::MatrixGroup(m, n, R)(<m, n>, List, <Banded>)
Dom::MatrixGroup(m, n, R) creates a domain which represents the Abelian group of m×n matrices over the component ring R, i.e., it is a domain of category Cat::AbelianGroup.
The domain Dom::ExpressionField() is used as the component ring for the matrices if the optional parameter R is not given.
For matrices of a domain created by Dom::MatrixGroup(m, n, R), matrix arithmetic is implemented by overloading the standard arithmetical operators +, -, *, / and ^. All functions of the linalg package dealing with matrices can be applied.
Dom::MatrixGroup(m, n, R) has the domain Dom::Matrix(R) as its super domain, i.e., it inherits each method which is defined by Dom::Matrix(R) and not re-implemented by Dom::MatrixGroup(m, n, R).
Methods described below are implemented by Dom::MatrixGroup.
The domain Dom::Matrix(R) represents matrices over R of arbitrary size, and it therefore does not have any algebraic structure (except of being a set of matrices).
The domain Dom::SquareMatrix(n, R) represents the ring of n×n matrices over R.
Dom::MatrixGroup(m, n, R)(Array) and Dom::MatrixGroup(m, n, R)(Matrix) create a new matrix formed by the entries of Array and Matrix, respectively.
The components of Array and Matrix, respectively, are converted into elements of the domain R. An error message is issued if one of these conversions fails.
The call Dom::MatrixGroup(m, n, R)( m , n ) returns the m×n zero matrix. Note that the m×n zero matrix can also be found in the entry "zero" (see below).
Dom::MatrixGroup(m, n, R)( m , n List) creates an m×n matrix with components taken from the list List.
This call is only allowed for m×1 or 1 ×n matrices, i.e., if either m or n is equal to one.
If the list has too few entries, the remaining components of the matrix are set to zero.
The entries of the list are converted into elements of the domain R. An error message is issued if one of these conversions fails.
Dom::MatrixGroup(m, n, R)( m , n ListOfRows) creates an m×n matrix with components taken from the nested list ListOfRows. Each inner list corresponds to a row of the matrix.
If an inner list has less than n entries, the remaining components in the corresponding row of the matrix are set to zero. If there are less than m inner lists, the remaining lower rows of the matrix are filled with zeroes.
The entries of the inner lists are coerced into elements of the domain R. An error message is issued if one of these conversions fails.
It might be a good idea first to create a two-dimensional array from that list before calling Dom::MatrixGroup(m, n, R). This is due to the fact that creating a matrix from an array is the fastest way one can achieve. However, in this case the sublists must have the same number of elements.
Dom::MatrixGroup(m, n, R)( m , n f) returns the matrix whose (i, j)th component is the value of the function call f(i, j). The row index i ranges from 1 to m and the column index j from 1 to n.
The function values are coerced into elements of the domain R. An error message is issued if one of these conversions fails.
A lot of examples can be found on the help page of the domain constructor Dom::Matrix, and most of them are also examples for working with domains created by Dom::MatrixGroup. This example only highlights some differences with respect to working with matrices of the domain Dom::Matrix(R).
The following command defines the abelian group of 3 ×4 matrices over the rationals:
MatGQ := Dom::MatrixGroup(3, 4, Dom::Rational)
MatGQ::hasProp(Cat::AbelianGroup), MatGQ::hasProp(Cat::Ring)
MatGQ is a commutative group with respect to the addition of matrices. The unit of this group is the 3 ×4 zero matrix:
MatGQ::zero
Note that some operations defined by the domain MatGQ return matrices which are no longer elements of the matrix group. They return matrices of the domain Dom::Matrix(Dom::Rational), the super-domain of MatGQ.
For example, if we define the matrix:
A := MatGQ([[1, 2, 1, 2], [-5, 3], [2, 1/3, 0, 1]])
and delete its third column, we get the matrix:
MatGQ::delCol(A, 3)
which is of the domain type:
domtype(%)
As another example we create the 3×3 identity matrix using the method "identity" of our domain:
E3 := MatGQ::identity(3)
This is also a matrix of the domain Dom::Matrix(Dom::Rational):
domtype(E3)
If we concatenate E3 to the right of the matrix A defined above, we get the 3 ×7 matrix:
B := A . E3
which is of the domain type Dom::Matrix(Dom::Rational):
domtype(B)
We can convert a matrix from a domain created with Dom::MatrixGroup into or from another matrix domain, as shown next:
MatGR := Dom::MatrixGroup(2, 3, Dom::Real): MatC := Dom::Matrix(Dom::Complex):
A := MatGR((i, j) -> i*j)
To convert A into a matrix of the domain MatC, enter:
coerce(A, MatC)
domtype(%)
The conversion is done component-wise. For example, we define the following matrix:
B := MatC([[0, 1, 0], [exp(I), 0, 1]])
The matrix B has one complex component and therefore cannot be converted into the domain MatGR:
coerce(B, MatGR)
Note: The system function coerce uses the methods "convert" and "convert_to" implemented by any domain created with Dom::MatrixGroup and Dom::Matrix.
m, n |
Positive integers (matrix dimension) |
R |
A commutative ring, i.e., a domain of category Cat::CommutativeRing; the default is Dom::ExpressionField() |
Array |
An m×n array |
Matrix |
An m×n matrix, i.e., an element of a domain of category Cat::Matrix |
List |
A list of matrix components |
ListOfRows |
A list of at most m rows; each row is a list of at most n matrix components |
f |
A function or a functional expression with two parameters (the row and column index) |
g |
A function or a functional expression with one parameter (the row index) |
Diagonal |
Create a diagonal matrix With the option Diagonal, diagonal matrices can be created with diagonal elements taken from a list, or computed by a function. Dom::MatrixGroup(m, n, R)( m , n List, Diagonal) creates the m×n diagonal matrix whose diagonal elements are the entries of List. List must have at most min(m, n) entries. If it has fewer elements, then the remaining diagonal elements are set to zero. The entries of List are coerced into elements of the domain R. An error message is issued if one of these conversions fails. Dom::MatrixGroup(m, n, R)( m , n g, Diagonal) returns the matrix whose ith diagonal element is g(i, i), where the index i runs from 1 to min(m, n). The function values are coerced into elements of the domain R. An error message is issued if one of these conversions fails. |
Banded |
Create a banded Toeplitz matrix With the option Banded, banded matrices can be created. A banded matrix has all entries zero outside the main diagonal and some of the adjacent sub- and superdiagonals. Dom::MatrixGroup(m, n, R)( m , n List, Banded) creates an m×n banded Toeplitz matrix with the elements of List as entries. The number of entries of List must be odd, say 2 h + 1, and must not exceed n. The resulting matrix has bandwidth at most 2 h + 1. All elements of the main diagonal of the created matrix are initialized with the middle element of List. All elements of the ith subdiagonal are initialized with the (h + 1 - i)th element of List. All elements of the ith superdiagonal are initialized with the (h + 1 + i)th element of List. All entries on the remaining sub- and superdiagonals are set to zero. The entries of List are converted into elements of the domain R. An error message is issued if one of these conversions fails. |
"one" | is only defined if m is equal to n; in that case it defines the n×n identity matrix. |
"randomDimen" | is set to [m, n]. |
"zero" | is the m×n zero matrix. |