# Vectors and matrices

Vector and matrices are the fundamental building blocks of any linear algebra library. Their sizes can either be known at compile-time or only at run-time. In either case, they allow common operations (additions, multiplications, etc.) to be used through operator overloading.

`Matrix`

type#

The generic **nalgebra** represents both matrices and vectors of arbitrary dimensions (known at
compile-time or at runtime), using the same generic `Matrix`

structure. Vectors are just
matrices with only one column. But in practice, users will generally rely on some more
convenient aliased for this generic matrix type:

`SMatrix<T, R, C>`

are statically-sized matrices with R rows and C columns, e.g.,`SMatrix<f32, 24, 10>`

.`SVector<T, D>`

is a statically-sized column vector with`D`

rows, e.g.,`SVector<f32, 10>`

.`DVector<T>`

and`DMatrix<T>`

: are respectively a dynamically-sized column vector and a dynamically-sized matrix.

Aliases exist for statically-sized vectors and matrices with small dimensions:

`Vector1<T>`

..`Vector6<T>`

: are column vectors of dimension 1 to 6.`Matrix1<T>`

..`Matrix6<T>`

: are square matrices of dimension 1x1 to 6x6.- Rectangular matrices have the form
`MatrixIxJ<T>`

where`I`

and`J`

are any value from 1 to 6, e.g.,`Matrix4x5<T>`

.

In its most general form, the `Matrix<T, R, C, S>`

type takes four type parameters:

`T`

: the scalar type, i.e., the type of the matrix components. Typical values are`f32`

or`f64`

.`R`

: a type characterizing the number of rows on this matrix.`C`

: a type characterizing the number of columns on this matrix.`S`

: the buffer that contains all the matrix components.

The type parameters `R`

and `C`

completely determine whether or not the matrix
shape is known at compile-time or only at run-time. They can have two kinds of
values:

**Constant integers:**Represented by the type`Const<const D: usize>`

. For example,`Matrix<T, Const<2>, Const<4>, S>`

represents a matrix with 2 rows and 4 columns.**The**using`Dynamic`

type:`Dynamic`

instead of a`Const<N>`

integer indicates that the corresponding matrix dimension is not known at compile-time. For example,`Matrix<T, Dynamic, Dynamic, S>`

has a number of rows and number of columns that can only be known at runtime. Another typical example is a dynamically-sized column vector:`Matrix<T, Dynamic, Const<1>, S>`

.

##### note

For all constant integers in `[0; 127]`

, you may use `U1, U2, ..., U127`

which
are shorter aliases for `Const<1>, Const<2>, ..., Const<127>`

.

Internally, dynamically- and statically-sized matrices do not use the same data
storage type. While the former is always allocated on the heap using a `Vec`

,
the latter prefers static allocation indirectly using a column-major 2D array
`[[T; R]; C]`

for better performances.

## #

Matrix constructionAll matrices and vectors with shapes known at compile-time can be created from
the values of their components given in conventional mathematical notation,
i.e., row-by-rows, using the usual `::new`

method:

Other construction methods include:

Method | Description |
---|---|

`::from_rows(...)` | Creates a matrix filled with the given array of rows. Panics if any two rows provided do not have the same size. |

`::from_columns(...)` | Creates a matrix filled with the given array of columns. Panics if any two columns provided do not have the same size. |

`::from_diagonal(...)` | Creates a diagonal matrix with its diagonal equal to the provided vector. All off-diagonal elements are set to `0` . |

`::repeat(...)` | Creates a matrix filled with the given element (same as `::from_element(...)` ). |

`::from_element(...)` | Creates a matrix filled with the given element (same as `::repeat(...)` ). |

`::from_iterator(...)` | Creates a matrix filled with the content of the given iterator. The iterator must provide the matrix components in column-major order. |

`::from_row_slice(...)` | Creates a matrix filled with the content of the given slice. Elements of the slice are provided in row-major order (which is the usual mathematical notation.) |

`::from_column_slice(...)` | Creates a matrix filled with the content of the given slice. Elements of the slice are provided in column-major order. |

`::from_vec(...)` | Creates a matrix filled with the content of the given `Vec` . Elements of the vec are provided in column-major order. Constructing a dynamically-sized matrix this way consumes the vec to avoid allocations. |

`::from_fn(...)` | Creates a matrix filled with the values returned by the given closure of type `FnMut(usize, usize) -> T` . This closure is called exactly once per matrix component and is given as parameter each matrix component's 0-based indices. |

`::identity(...)` | Creates a matrix with `1` on its diagonal and `0` elsewhere. If the matrix to be constructed is not square, only the largest square submatrix formed by its first rows and columns is set to the identity matrix. All the other components are `0` . |

`::from_diagonal_element(...)` | Creates a matrix with its diagonal filled with the given element and `0` elsewhere. If the matrix to be constructed is not square, only the largest square submatrix formed by its first rows and columns is set to the identity matrix. All the other components are set to `0` . |

`::new_random(...)` | Creates a matrix with all its components initialized at random using the default random generator of the `rand` crate, i.e., the `rand::random()` function. |

Matrices with sizes known at compile-time implement some construction traits from the `num`

crate at well:

Trait method | Description |
---|---|

`Zero::zero()` | Creates a matrix filled with zeros. |

`One::one()` | Creates a matrix with a diagonal set to `1` and off-diagonal elements set to `0` . |

`Bounded::min_value()` | Creates a matrix filled with the minimal value of the matrix scalar type. |

`Bounded::max_value()` | Creates a matrix filled with the maximal value of the matrix scalar type. |

Column vectors (which are just matrices with a single column) with low dimensions from 1 to 6 have additional constructors:

`::x()`

,`::y()`

, and`::z()`

create a vector with, respectively, the first, second, or third coordinate set to`1`

and all others to`0`

.`::a()`

,`::b()`

, and`::c()`

create a vector with, respectively, the fourth, fifth, or sixth coordinate set to`1`

and all others to`0`

.

Adding a `_axis`

suffix to those constructors, e.g., `::y_axis()`

, will create
a unit vector wrapped into the `Unit`

structure. For example,
`Vector2::y_axis()`

will create a `Unit<Vector2<T>>`

with its the second
component of the underlying vector set to `1`

.

## #

Matrix operationsOperations between two matrices like addition, division, and multiplication, require both matrices to have compatible shapes. In particular:

- Addition require both matrices to have the same number of rows
**and**the same number of columns. - Multiplication and division requires the matrix on the left-hand-side to have as many columns as the number of rows of the matrix on the right-hand-side.

Those restrictions are either checked at compile-time or at runtime, depending on the inputs types. In particular, if the matrix dimensions to be checked are const integers then the check is performed at compile-time. The following shows an example of compilation error for attempting to multiply a 2x3 matrix with a 4x4 matrix:

If at least one matrix dimension to be checked is `Dynamic`

then the check is
performed at run-time and panics in case of mismatch. The following example
shows the run-time error for attempting to multiply a statically-sized 2x3
matrix with a dynamically-sized 4x4 matrix:

The return type of a matrix operation is automatically deduced from the matrix dimensions:

- If both matrices have dimensions known at compile-time then the result also has dimensions known at compile-time.
- If both matrices have dimensions known at run-time only then the result also has dimensions known at run-time.
- If one matrix has dimensions known at run-time and the other has dimensions
known at compile-time then the result will have dimensions known at
compile-time if they can be statically deduced from the arguments. For example,
adding a
`Matrix2x3`

to a`DMatrix`

will return a`Matrix2x3`

. However, multiplying a`Matrix2x3`

to a`DMatrix`

will return a matrix with one dimension known at compile-time, and a second one known at run-time, i.e.,`Matrix<T, U2, Dynamic, S>`

(where`T`

and`S`

are some types not detailed here). Indeed, the number of rows can be deduced from the first argument but the number of columns depends on the run-time value stored by the second argument.

## #

Matrix element modificationIt is possible to modify elements of a matrix when it is mutable. The following examples show in particular how to:

- Modify a single element.
- Modify an entire row/column.

Because types like `Vector3`

are column vectors, i.e., matrices with dimension 3×1, they can only be used with
`.set_column(...)`

. For setting a row with `.set_row(...)`

it is necessary to use a row vector. For example `RowVector4`

is a 4D row vector, i.e., a matrix with dimensions 4×1.

## #

Matrix viewsMatrix (and vector) views allow you to take a reference to a part of any matrix. A view does not perform any copy, move, or allocation of the original matrix data. Instead, it stores a pointer to that data together with some metadata about the view size and strides. Note that taking a view of a matrix view is allowed!

Because a matrix view is also just a special case of the generic `Matrix<T, R, C, S>`

type, it can usually be used just like a plain, non-view matrix besides three exceptions:

- Methods that require a
`&mut self`

cannot be called on non-mutable views. - Matrix views cannot be created out of thin air using the methods shown in the Matrix construction section. One must already have an instance of a matrix or another view and use one of the dedicated methods shown thereafter.
- Assignment operators do not work on any kind of view, i.e., one cannot
write
`a *= b`

even if`a`

is a mutable matrix view.

There are three variations of matrix view methods. Mutable views follow the
same semantics, except that the method names end with `_mut`

:

**"Fixed" views:**views with numbers of rows and columns known at compile-time. The name of the corresponding view methods usually start with the prefix`fixed_`

.

Method | Description |
---|---|

`.row(i)` | Reference to the i-th row of `self` . |

`.column(i)` | Reference to the i-th column of `self` . |

`.fixed_rows::<D>(i)` | Reference to the submatrix with `D` consecutive rows of `self` , starting with the i-th. `D` must be an integer. |

`.fixed_columns::<D>(i)` | Reference to the submatrix with `D` consecutive columns of `self` , starting with the i-th. `D` must be an integer. |

`.fixed_view::<R, C>(irow, icol)` | Reference to the submatrix with `R` consecutive rows and `C` consecutive columns, starting with the `irow` -th row and `icol` -th column. `R` and `C` are integers. |

**"Dynamic" views:**views with numbers of rows and columns known at run-time only.

Method | Description |
---|---|

`.rows(i, size)` | Reference to `size` rows of `self` , starting with the i-th. |

`.columns(i, size)` | Reference to `size` columns of `self` , starting with the i-th. |

`.view(start, shape)` | Reference to the submatrix with `shape.0` rows and `shape.1` columns, starting with the `start.0` -th row and `start.1` -th column. `start` and `shape` are both tuples. |

**Views with strides:**fixed or dynamic views that reference non-consecutive (but regularly spaced) rows and columns of the original matrix. The name of the corresponding view methods end with`_with_step`

.

Method | Description |
---|---|

`.fixed_rows_with_step::<D>(i, step)` | Reference to `D` non-consecutive rows of `self` , starting with the i-th. `step` rows of `self` are skipped between each referenced row. |

`.fixed_columns_with_step::<D>(i, step)` | Reference to `D` non-consecutive columns of `self` , starting with the i-th. `step` columns of `self` are skipped between each referenced column. |

`.fixed_view_with_steps::<R, C>(start, step)` | Reference to `R` and `C` non-consecutive rows and columns, starting with the component `(start.0, start.1)` . `step.0` (resp. `step.1` ) rows (resp. columns) are skipped between each referenced row (resp. column). |

`.rows_with_step(i, size, step)` | Reference to `size` rows of `self` , starting with the i-th. `step` rows are skipped between each referenced row. |

`.columns_with_step(i, size, step)` | Reference to `size` columns of `self` , starting with the i-th. `step` columns are skipped between each reference column. |

`.view_with_steps(start, shape, steps)` | Reference to `shape.0` rows and `shape.1` columns, starting with the `(start.0, start.1)` -th component. `step.0` (resp. `step.1` ) rows (resp. columns) are skipped between each referenced row (resp. column). |

Note that the method `.clone_owned()`

may be used to create a plain matrix from
a view, i.e., actually copying the referenced components into a new matrix
structure that owns its data. Whether or not the result of this cloning is a
dynamically- or statically-sized matrix depends on the kind of view.
Fixed views will yield a statically-sized matrix while dynamic views yield
a dynamically-sized matrix.

## #

Matrix resizingThe number of rows or columns of a matrix can be modified by adding or removing some of them. Similarly to views, two variants exist:

**"Fixed resizing"**where the number of rows or columns to be removed or inserted are known at compile-time. This allows the compiler to output a statically-sized matrix when the input is also statically-sized.

Method | Description |
---|---|

`.remove_row(i)` | Removes the i-th row. |

`.remove_column(i)` | Removes the i-th column. |

`.remove_fixed_rows::<D>(i)` | Removes `D` consecutive rows, starting with the i-th. |

`.remove_fixed_columns::<D>(i)` | Removes `D` consecutive columns, starting with the i-th. |

`.insert_row(i, val)` | Adds one row filled with `val` at the i-th row position. |

`.insert_column(i, val)` | Adds one column filled with `val` at the i-th row position. |

`.insert_fixed_rows::<D>(i, val)` | Adds `D` consecutive rows filled with `val` starting at the i-th row position. |

`.insert_fixed_columns::<D>(i, val)` | Adds `D` consecutive columns filled with `val` starting at the i-th column position. |

`.fixed_resize::<R2, C2>(val)` | Resizes the matrix so that it contains `R2` rows and `C2` columns. Components are copied such that `result[(i, j)] == input[(i, j)]` . If the result matrix has more rows or more columns, then the extra components are initialized to `val` . |

**"Dynamic resizing"**where the number of rows or columns to be removed or inserted are not known at compile-time. The result matrix will always be dynamically-sized (the affected dimension-related type parameter of`Matrix<...>`

is set to`Dynamic`

).

Method | Description |
---|---|

`.remove_rows(i, n)` | Removes `n` rows, starting with the i-th. |

`.remove_columns(i, n)` | Removes `n` columns, starting with the i-th. |

`.insert_rows(i, n, val)` | Inserts `n` rows filled with `val` starting at the i-th row position. |

`.insert_columns(i, n, val)` | Inserts `n` columns filled with `val` starting at the i-th row position. |

`.resize(new_nrows, new_ncols, val)` | Resizes the matrix so that it contains `new_nrows` rows and `new_ncols` columns. Components are copied such that `result[(i, j)] == input[(i, j)]` . If the result matrix has more rows or more columns, then the extra components are initialized to `val` . |

The implicit `self`

argument of those methods is always consumed in order to
re-use the input data storage to construct the output. Fixed resizing should be
preferred whenever the number of rows/columns to be inserted or removed is
known at compile-time.

It is strongly recommended to use fixed resizing whenever possible, especially when the matrix being resize has a size known at compile-time (and is thus statically allocated). Indeed, dynamic resizing will produce heap-allocated results because the size of the output matrix cannot be deduced at compile-time.