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### 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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