# Dom::MatrixGroup

The Abelian group of matrices

## Syntax

### Domain Creation

`Dom::MatrixGroup(`m`, `n`, <`R`>)`

### Element Creation

`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>)`

## Description

### Domain Creation

`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.

### Element Creation

`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.

## Superdomain

`Dom::Matrix``(R)`

## Axioms

If `R` has `Ax::canonicalRep`, then `Ax::canonicalRep`.

## Categories

`Cat::Matrix``(R)`, `Cat::AbelianGroup`

## Examples

### Example 1

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)`

### Example 2

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`.

## Parameters

 `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)

## Options

 `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.

## Entries

 "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.

expand all

## Mathematical Methods

### `evalp` — Evaluating matrices of polynomials at a certain point

`evalp(A, x = a, …)`

This method is only defined if `R` is a polynomial ring of category `Cat::Polynomial`.

This method overloads the function `evalp` for matrices, i.e., one may use it in the form `evalp(A, x = a)`.

### `identity` — Identity matrix

`identity(k)`

 Note:   The matrix returned is of the domain `Dom::Matrix(R)`, if or if .

### `matdim` — Matrix dimension

`matdim(A)`

### `random` — Random matrix generation

`random()`

The components of the random matrix are randomly generated with the method `"random"` of the component ring `R`.

## Access Methods

### `_concat` — Horizontally concatenation of matrices

`_concat(A, B, …)`

An error message is issued if the given matrices do not have the same number of rows.

 Note:   The returned matrix is of the domain `Dom::Matrix(R)`.

This method overloads the function `_concat` for matrices, i.e., one may use it in the form `A . B . ...`, or in functional notation: ```_concat(A, B, ...)```.

### `_index` — Matrix indexing

`_index(A, i, j)`

```_index(A, r1 .. r2, c1 .. c2)```

`_index(A, i)`

`_index(A, i1 .. i2)`

If `i` and `j` are not integers, then the call of this method returns in its symbolic form (of type `"_index"`) with evaluated arguments.

Otherwise an error message is given, if `i` and `j` are not valid row and column indices, respectively.

 Note:   Note that the system function `context` is used to evaluate the entry in the context of the calling environment.

This method overloads the function `_index` for matrices, i.e., one may use it in the form `A[i, j]` or in functional notation: ```_index(A, i, j)```.

Returns the submatrix of `A`, created by the rows of `A` with indices from `r1` to `r2` and the columns of `A` with indices from `c1` to `c2`.

 Note:   The submatrix is of the domain `Dom::Matrix(R)`.

This method returns the ith entry of `A`.

If `i` is not an integer, then the call of this method returns in its symbolic form (of type `"_index"`) with evaluated arguments.

Otherwise an error message is given, if `i` is less than one or greater than the dimension of `v`.

This call is only allowed for 1×n or m×1 matrices, i.e., either `m` or `n` must be equal to one. Otherwise an error message is issued.

 Note:   Note that the system function `context` is used to evaluate the entry in the context of the calling environment.

This method returns the subvector of `A`, formed by the entries with index `i1` to `i2` (see also the method `"op"`).

This call is only allowed for 1×n or m×1 matrices, i.e., either `m` or `n` must be equal to one. Otherwise an error message is issued.

This method overloads the function `_index` for matrices, i.e., one may use it in the form `A[i,j]`, `A[r1..r2,c1..c2]`, `A[i]` or `A[i1..i2]`, respectively, or in functional notation: `_index(A, ...)`.

### `concatMatrix` — Horizontally concatenation of matrices

```concatMatrix(A, B, …)```

### `col` — Extracting a column

`col(A, c)`

An error message is issued if `c` is less than one or greater than `n`.

### `delCol` — Deleting a column

`delCol(A, c)`

`NIL` is returned if `A` only consists of one column.

 Note:   The returned matrix is of the domain `Dom::Matrix(R)`.

An error message is issued if `c` is less than one or greater than `n`.

### `delRow` — Deleting a row

`delRow(A, r)`

`NIL` is returned if `A` only consists of one row.

 Note:   The returned matrix is of the domain `Dom::Matrix(R)`.

An error message is issued if `r` is less than one or greater than `m`.

### `row` — Extracting a row

`row(A, r)`

An error message is issued if `r` is less than one or greater than `m`.

### `stackMatrix` — Concatenating of matrices vertically

`stackMatrix(A, B, …)`

An error message is issued if the given matrices do not have the same number of columns.

 Note:   The matrix returned is of the domain `Dom::Matrix(R)`.

## Conversion Methods

### `convert` — Conversion into a matrix

`convert(x)`

`FAIL` is returned if the conversion fails.

`x` may either be an m×n array, or an m×n matrix of category `Cat::Matrix`.

`x` can also be a list. See the parameter `List` and `ListOfRows` in "Creating Elements" above for admissible values of `x`.

The entries of `x` must be convertable into elements of the domain `R`, otherwise `FAIL` is returned.