# Matrices

Singular internally allows for matrices over polynomial rings to be created extremely efficiently from ideals and modules (often without copying data). This allows for introspection of modules and operations that can be expressed in terms of matrices (e.g. composition of $R$-module homomorphisms) to be computed, at a low level.

The default matrix type in Singular.jl is the `smatrix`

type.

Matrix objects have a parent object which represents the space of matrices they belong to, the data for which is given by the polynomial ring $R$ over which the matrices are defined, and the number of rows and columns of the matrices in the space.

The types of matrices and associated parent objects are given in the following table according to the library providing them.

Library | Element type | Parent type |
---|---|---|

Singular | `smatrix{T}` | `Singular.matrix_space{T}` |

These types are parameterised by the type of elements in the polynomial ring $R$ over which the matrices are defined.

All matrix types belong directly to the abstract type `SetElem`

and all the matrix space parent object types belong to the abstract type `Set`

.

## Matrix functionality

Singular.jl matrices implement standard operations one would expect. These include:

- Operations common to all AbstractAlgebra objects, such as
`parent`

,`base_ring`

,`elem_type`

,`parent_type`

,`parent`

,`deepcopy`

, etc.

The following parts of the Matrix interface from AbstractAlgebra are also implemented:

- construction:
`identity_matrix`

,`identity_matrix`

- arithmetic operations:
`+`

,`-`

,`*`

- comparison:
`==`

- manipulation:
`number_of_rows`

,`number_of_columns`

,`getindex`

,`setindex!`

,`transpose`

,`iszero`

**Examples**

```
julia> R, (x, y, u, v, w) = Singular.polynomial_ring(Singular.QQ, ["x", "y", "u", "v", "w"])
(Singular polynomial ring (QQ),(x,y,u,v,w),(dp(5),C), spoly{n_Q}[x, y, u, v, w])
julia> identity_matrix(R, 4)
[1, 0, 0, 0
0, 1, 0, 0
0, 0, 1, 0
0, 0, 0, 1]
julia> zero_matrix(R, 3, 8)
[0, 0, 0, 0, 0, 0, 0, 0
0, 0, 0, 0, 0, 0, 0, 0
0, 0, 0, 0, 0, 0, 0, 0]
julia> M = identity_matrix(R, 4)
[1, 0, 0, 0
0, 1, 0, 0
0, 0, 1, 0
0, 0, 0, 1]
julia> number_of_rows(M)
4
julia> number_of_columns(M)
4
julia> iszero(M)
false
julia> M[3, 4] = x*y + 5*u*w
x*y + 5*u*w
julia> N = transpose(M)
[1, 0, 0, 0
0, 1, 0, 0
0, 0, 1, 0
0, 0, x*y + 5*u*w, 1]
julia> N[4, 3]
x*y + 5*u*w
```