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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