GAP (LinearAlgebraForCAP) - Chapter 1: Category of Matrices

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1 Category of Matrices
1.1 Constructors
1.1-1 MatrixCategory
1.1-2 MatrixCategoryAsCategoryOfRows
1.1-3 VectorSpaceMorphism
1.1-4 VectorSpaceObject
1.1-5 MatrixCategoryObject
1.2 Attributes
1.2-1 UnderlyingFieldForHomalg
1.2-2 UnderlyingMatrix
1.2-3 UnderlyingFieldForHomalg
1.2-4 Dimension
1.3 GAP Categories
1.3-1 IsVectorSpaceMorphism
1.3-2 IsVectorSpaceObject

### 1 Category of Matrices

#### 1.1 Constructors

##### 1.1-1 MatrixCategory

`‣ MatrixCategory`( F ) | ( attribute )

Returns: a category

The argument is a homalg field \(F\). The output is the matrix category over \(F\). Objects in this category are non-negative integers. Morphisms from a non-negative integer \(m\) to a non-negative integer \(n\) are given by \(m \times n\) matrices.

##### 1.1-2 MatrixCategoryAsCategoryOfRows

`‣ MatrixCategoryAsCategoryOfRows`( F ) | ( operation )

Returns: a category

The argument is a homalg field \(F\). The output is the matrix category over \(F\), constructed internally as a wrapper category of the `CategoryOfRows` of \(F\). Only available if the package `FreydCategoriesForCAP` is available.

##### 1.1-3 VectorSpaceMorphism

`‣ VectorSpaceMorphism`( S, M, R ) | ( operation )

Returns: a morphism in \(\mathrm{Hom}(S,R)\)

The arguments are an object \(S\) in the category of matrices over a homalg field \(F\), a homalg matrix \(M\) over \(F\), and another object \(R\) in the category of matrices over \(F\). The output is the morphism \(S \rightarrow R\) in the category of matrices over \(F\) whose underlying matrix is given by \(M\).

##### 1.1-4 VectorSpaceObject

`‣ VectorSpaceObject`( d, F ) | ( operation )

Returns: an object

The arguments are a non-negative integer \(d\) and a homalg field \(F\). The output is an object in the category of matrices over \(F\) of dimension \(d\). This function delegates to `MatrixCategoryObject`.

##### 1.1-5 MatrixCategoryObject

`‣ MatrixCategoryObject`( cat, d ) | ( operation )

Returns: an object

The arguments are a matrix category \(cat\) over a field and a non-negative integer \(d\). The output is an object in \(cat\) of dimension \(d\).

#### 1.2 Attributes

##### 1.2-1 UnderlyingFieldForHomalg

`‣ UnderlyingFieldForHomalg`( alpha ) | ( attribute )

Returns: a homalg field

The argument is a morphism \(\alpha\) in the matrix category over a homalg field \(F\). The output is the field \(F\).

##### 1.2-2 UnderlyingMatrix

`‣ UnderlyingMatrix`( alpha ) | ( attribute )

Returns: a homalg matrix

The argument is a morphism \(\alpha\) in a matrix category. The output is its underlying matrix \(M\).

##### 1.2-3 UnderlyingFieldForHomalg

`‣ UnderlyingFieldForHomalg`( A ) | ( attribute )

Returns: a homalg field

The argument is an object \(A\) in the matrix category over a homalg field \(F\). The output is the field \(F\).

##### 1.2-4 Dimension

`‣ Dimension`( A ) | ( attribute )

Returns: a non-negative integer

The argument is an object \(A\) in a matrix category. The output is the dimension of \(A\).

#### 1.3 GAP Categories

##### 1.3-1 IsVectorSpaceMorphism

`‣ IsVectorSpaceMorphism`( object ) | ( filter )

Returns: `true` or `false`

The GAP category of morphisms in the category of matrices of a field \(F\).

##### 1.3-2 IsVectorSpaceObject

`‣ IsVectorSpaceObject`( object ) | ( filter )

Returns: `true` or `false`

The GAP category of objects in the category of matrices of a field \(F\).

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