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Matrix product. This can compute any combination of sparse and dense multiplication.

>>> V2 (V3 1 2 3) (V3 4 5 6) !*! V3 (V2 1 2) (V2 3 4) (V2 4 5)
V2 (V2 19 25) (V2 43 58)

>>> V2 (IntMap.fromList [(1,2)]) (IntMap.fromList [(2,3)]) !*! IntMap.fromList [(1,V3 0 0 1), (2, V3 0 0 5)]
V2 (V3 0 0 2) (V3 0 0 15)

Entry-wise matrix addition.

>>> V2 (V3 1 2 3) (V3 4 5 6) !+! V2 (V3 7 8 9) (V3 1 2 3)
V2 (V3 8 10 12) (V3 5 7 9)

Entry-wise matrix subtraction.

>>> V2 (V3 1 2 3) (V3 4 5 6) !-! V2 (V3 7 8 9) (V3 1 2 3)
V2 (V3 (-6) (-6) (-6)) (V3 3 3 3)

Matrix * column vector

>>> V2 (V3 1 2 3) (V3 4 5 6) !* V3 7 8 9
V2 50 122

Row vector * matrix

>>> V2 1 2 *! V2 (V3 3 4 5) (V3 6 7 8)
V3 15 18 21

Matrix-scalar product

>>> V2 (V2 1 2) (V2 3 4) !!* 5
V2 (V2 5 10) (V2 15 20)

Scalar-matrix product

>>> 5 *!! V2 (V2 1 2) (V2 3 4)
V2 (V2 5 10) (V2 15 20)

Matrix-scalar division

This is a generalization of inside to work over any corepresentable Functor.

column :: Representable f => Lens s t a b -> Lens (f s) (f t) (f a) (f b)

In practice it is used to access a column of a matrix.

>>> V2 (V3 1 2 3) (V3 4 5 6) ^._x
V3 1 2 3

>>> V2 (V3 1 2 3) (V3 4 5 6) ^.column _x
V2 1 4

Hermitian conjugate or conjugate transpose

>>> adjoint (V2 (V2 (1 :+ 2) (3 :+ 4)) (V2 (5 :+ 6) (7 :+ 8)))
V2 (V2 (1.0 :+ (-2.0)) (5.0 :+ (-6.0))) (V2 (3.0 :+ (-4.0)) (7.0 :+ (-8.0)))

A 2x2 matrix with row-major representation

A 2x3 matrix with row-major representation

A 2x4 matrix with row-major representation

A 3x2 matrix with row-major representation

A 3x3 matrix with row-major representation

A 3x4 matrix with row-major representation

A 4x2 matrix with row-major representation

A 4x3 matrix with row-major representation

A 4x4 matrix with row-major representation

Convert a 3x3 matrix to a 4x4 matrix extending it with 0's in the new row and column.

Convert from a 4x3 matrix to a 4x4 matrix, extending it with the [ 0 0 0 1 ] column vector

2x2 matrix determinant.

>>> det22 (V2 (V2 a b) (V2 c d))
a * d - b * c

3x3 matrix determinant.

>>> det33 (V3 (V3 a b c) (V3 d e f) (V3 g h i))
a * (e * i - f * h) - d * (b * i - c * h) + g * (b * f - c * e)

4x4 matrix determinant.

2x2 matrix inverse.

>>> inv22 $ V2 (V2 1 2) (V2 3 4)
V2 (V2 (-2.0) 1.0) (V2 1.5 (-0.5))

3x3 matrix inverse.

>>> inv33 $ V3 (V3 1 2 4) (V3 4 2 2) (V3 1 1 1)
V3 (V3 0.0 0.5 (-1.0)) (V3 (-0.5) (-0.75) 3.5) (V3 0.5 0.25 (-1.5))

4x4 matrix inverse.

The identity matrix for any dimension vector.

>>> identity :: M44 Int
V4 (V4 1 0 0 0) (V4 0 1 0 0) (V4 0 0 1 0) (V4 0 0 0 1)
>>> identity :: V3 (V3 Int)
V3 (V3 1 0 0) (V3 0 1 0) (V3 0 0 1)

Extract the translation vector (first three entries of the last column) from a 3x4 or 4x4 matrix.

transpose is just an alias for distribute

transpose (V3 (V2 1 2) (V2 3 4) (V2 5 6))

V2 (V3 1 3 5) (V3 2 4 6)

Build a rotation matrix from a unit Quaternion.

Build a transformation matrix from a rotation expressed as a Quaternion and a translation vector.

Build a transformation matrix from a rotation matrix and a translation vector.

Extract a 2x2 matrix from a matrix of higher dimensions by dropping excess rows and columns.

Extract a 2x3 matrix from a matrix of higher dimensions by dropping excess rows and columns.

Extract a 2x4 matrix from a matrix of higher dimensions by dropping excess rows and columns.

Extract a 3x2 matrix from a matrix of higher dimensions by dropping excess rows and columns.

Extract a 3x3 matrix from a matrix of higher dimensions by dropping excess rows and columns.

Extract a 3x4 matrix from a matrix of higher dimensions by dropping excess rows and columns.

Extract a 4x2 matrix from a matrix of higher dimensions by dropping excess rows and columns.

Extract a 4x3 matrix from a matrix of higher dimensions by dropping excess rows and columns.

Extract a 4x4 matrix from a matrix of higher dimensions by dropping excess rows and columns.

Compute the (L, U) decomposition of a square matrix using Crout's algorithm. The Index of the vectors must be Integral.

Compute the (L, U) decomposition of a square matrix using Crout's algorithm, using the vector's Finite instance to provide an index.

Solve a linear system with a lower-triangular matrix of coefficients with forwards substitution.

Solve a linear system with a lower-triangular matrix of coefficients with forwards substitution, using the vector's Finite instance to provide an index.

Solve a linear system with an upper-triangular matrix of coefficients with backwards substitution.

Solve a linear system with an upper-triangular matrix of coefficients with backwards substitution, using the vector's Finite instance to provide an index.

Solve a linear system with LU decomposition.

Solve a linear system with LU decomposition, using the vector's Finite instance to provide an index.

Invert a matrix with LU decomposition.

Invert a matrix with LU decomposition, using the vector's Finite instance to provide an index.

Compute the determinant of a matrix using LU decomposition.

Compute the determinant of a matrix using LU decomposition, using the vector's Finite instance to provide an index.