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jonathanfishbein1/linear-algebra/27.0.0

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Matrix

A module for Matrix

Types

type Matrix a
= Matrix (List (RowVector a))

Matrix type

type Solution a
= UniqueSolution (ColumnVector a)
| InfiniteSolutions
{ nullity : Int
, rank : Int
}

Type to represent result of Gauss-Jordan reduction if system is consistant

= Consistant (Solution a)
| Inconsistant String (Solution a)

Type to represent result of Gauss-Jordan reduction

= VectorDimension Int

Type to represent vector space such as R, R2, R3

type alias MatrixSpace a =
{ abelianGroup : AbelianGroup (Matrix a)
, matrixScalarMultiplication : a -> Matrix a -> Matrix a
}

Type to represent a Vector Space

Values

identity : Field a -> Int -> Matrix a

Create Identity Matrix with n dimension

zeros : Field a -> Int -> Int -> Matrix a

Create Matrix with m x n dimension filled with zeros

Constructors

Create a Matrix from a list of Column Vectors

Unitary Operations

scalarMultiplication : Field a -> a -> Matrix a -> Matrix a

Scalar multiplication over a Matrix

Transpose a Matrix

conjugate : Matrix (ComplexNumber number) -> Matrix (ComplexNumber number)

Take the complex conjugate of a Complex Numbered Matrix

adjoint : Matrix (ComplexNumber number) -> Matrix (ComplexNumber number)

Perform the adjoint operation on a Complex Numbered Matrix

subMatrix : Int -> Int -> Int -> Int -> Matrix a -> Matrix a

Calculate the submatrix given a starting and ending row and column index

nullSpace : Equality a -> InnerProductSpace a -> Matrix a -> Consistancy a

Calculate the null space of a matrix

leftNullSpace : Equality a -> InnerProductSpace a -> Matrix a -> Consistancy a

Calculate the left nullspace of a Matrix

getDiagonal : Matrix a -> Maybe (List a)

Get the diagonal of a Matrix

getDiagonalProduct : Field a -> Matrix a -> Maybe a

Get the Product of the diagonal of a Matrix

rank : InnerProductSpace a -> Matrix a -> Int

Calculate the rank of a Matrix

Binary Operations

add : Field a -> Matrix a -> Matrix a -> Matrix a

Add two Matrices together

subtract : Field a -> Matrix a -> Matrix a -> Matrix a

Subtract two Matrices

InnerProductSpace a
-> Matrix a
-> Result String (ColumnVector a)

Multiply a Vector by a Matrix

InnerProductSpace a
-> Matrix a
-> Matrix a
-> Result String (Matrix a)

Matrix Matrix multiplication

tensorProduct : Field a -> Matrix a -> Matrix a -> Matrix a

Calculate the tensor product of two Matricies

InnerProductSpace a
-> Matrix a
-> Matrix a
-> Result String (Matrix a)

Calculate the commuter of two Hermitian Matricies

Matrix Predicates and Properties

InnerProductSpace a
-> List (ColumnVector a)
-> Bool

Determine whether list of vectors are a basis for a space

areLinearlyIndependent : InnerProductSpace a -> List (ColumnVector a) -> Bool

Predicate to determine if a list of Vectors are linearly independent

VectorSpace a
-> List (ColumnVector a)
-> Result String Bool

Determine whether list of vectors spans a space

mDimension : Matrix a -> Int

Number of rows in Matrix

nDimension : Matrix a -> Int

Number of columns in Matrix

areRowEquivalent : VectorSpace a -> Matrix a -> Matrix a -> Bool

Predicate to determine if two matricies are row equivalent

all : (a -> Bool) -> Matrix a -> Bool

Predicate to determine if all values in the matric satisfy the given predicate

isOneToOne : InnerProductSpace a -> Matrix a -> Result String (Matrix a)

Determine whether a matirx is one to one

isOnto : InnerProductSpace a -> Matrix a -> Result String (Matrix a)

Determine whether a matirx is onto

Semigroup, Monoid, Group, Ring, Field, VectorSpace instances

realAdditionSemigroup : Semigroup (Matrix (Real Float))

Semigroup instance for Matrix under the addition operation with real values.

complexAdditionSemigroup : Semigroup (Matrix (ComplexNumber Float))

Semigroup instance for Matrix under the addition operation with complex values.

realAdditionCommutativeSemigroup : CommutativeSemigroup (Matrix (Real Float))

Commutative Semigroup instance for Matrix under the addition operation with real values.

complexAdditionCommutativeSemigroup : CommutativeSemigroup (Matrix (ComplexNumber Float))

Commutative Semigroup instance for Matrix under the addition operation with complex values.

realAdditionCommutativeMonoid : CommutativeMonoid (Matrix (Real Float))

Commutative Monoid instance for Matrix under the addition operation with real values.

complexAdditionCommutativeMonoid : CommutativeMonoid (Matrix (ComplexNumber Float))

Commutative Monoid instance for Matrix under the addition operation with complex values.

realMatrixAlgebra : MatrixAlgebra (Real Float)

Real Numbered Matrix Algebra

complexMatrixAlgebra : MatrixAlgebra (ComplexNumber Float)

Complex Numbered Matrix Algebra

realAdditionGroup : Group (Matrix (Real Float))

Group instance for Matrix under the addition operation with real values.

complexAdditionGroup : Group (Matrix (ComplexNumber Float))

Group instance for Matrix under the addition operation with complex values.

realAdditionAbelianGroup : AbelianGroup (Matrix (Real Float))

Abelian Group instance for Matrix under the addition operation with real values.

complexAdditionAbelianGroup : AbelianGroup (Matrix (ComplexNumber Float))

Abelian Group instance for Matrix under the addition operation with complex values.

realMatrixSpace : MatrixSpace (Real Float)

Real Numbered Vector Space for Matrix

complexMatrixSpace : MatrixSpace (ComplexNumber Float)

Complex Numbered Vector Space for Matrix

Monoid, Functor, Applicative, Monad, Foldable functions

Monoid empty for Matrix

Monoidally append Matricies together horizontally

concatVertical : Monoid (Matrix a)

Monoidally append Matricies together vertically

Append Matricies together horizontally

map : (a -> b) -> Matrix a -> Matrix b

Map over a Matrix

pure : a -> Matrix a

Applicative pure for Matrix

andMap : Matrix a -> Matrix (a -> b) -> Matrix b

Apply for Matrix

andThen : (a -> Matrix b) -> Matrix a -> Matrix b

Monad bind for Matrix

map2 : (a -> b -> c) -> Matrix a -> Matrix b -> Matrix c

Lift a binary function to work on Matrix

foldl : (a -> b -> b) -> b -> Matrix a -> b

Left fold over a Matrix

Equality

equal : (a -> a -> Bool) -> Equality (Matrix a)

Equal type for Matrix.

equalImplementation : (a -> a -> Bool) -> Matrix a -> Matrix a -> Bool

Compare two Matrices for equality

Matrix Forms

upperTriangle : VectorSpace a -> Matrix a -> Matrix a

Put a matrix into Upper Triangular Form

gaussianReduce : VectorSpace a -> Matrix a -> Matrix a

Gaussian Elimination

jordanReduce : VectorSpace a -> Matrix a -> Matrix a

Internal function for Jordan Elimination

gaussJordan : VectorSpace a -> Matrix a -> Matrix a

Function composition of Gaussian Elimination and Jordan Elimination

Solving

Equality a
-> InnerProductSpace a
-> Matrix a

Solve a system of linear equations using Gauss-Jordan elimination with explict column vector of constants

Manipulation

getAt : ( Int, Int ) -> Matrix a -> Maybe a

Get the value in a matrix at the specified row and column

setAt : ( Int, Int ) -> a -> Matrix a -> Matrix a

Set the value in a Matrix at the specified row and column

printRealMatrix : Matrix (Real Float) -> String

Print a Real matrix to a string

printComplexMatrix : Matrix (ComplexNumber Float) -> String

Print a Complex matrix to a string

readRealMatrix : String -> Result (List DeadEnd) (Matrix (Real Float))

Try to read a string into a Matrix

readComplexMatrix : String -> Result (List DeadEnd) (Matrix (ComplexNumber Float))

Try to read a string into a Matrix

Modules

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