oalg-base-3.0.0.0: Algebraic structures on oriented entities and limits as a tool kit to solve algebraic problems.

Copyright	(c) Erich Gut
License	BSD3
Maintainer	zerich.gut@gmail.com
Safe Haskell	None
Language	Haskell2010

OAlg.Entity.Matrix.Transformation

Contents

Row Trafo
Col Trafo
Diagonal Form

Description

elementary matrix transformations, i.e. operations of GLT on Matrix.

Synopsis

newtype RowTrafo x = RowTrafo (GLT x)
crTrafoRows :: Transformation x -> Col N (Row N x) -> Col N (Row N x)
newtype ColTrafo x = ColTrafo (GLT x)
crTrafoCols :: Col N (Row N x) -> Transformation x -> Col N (Row N x)
data DiagonalForm k = DiagonalForm [k] (RowTrafo k) (ColTrafo k)
dgfMatrix :: Distributive k => DiagonalForm k -> Matrix k
newtype DiagonalFormStrictPositive k = DiagonalFormStrictPositive (DiagonalForm k)

Row Trafo

newtype RowTrafo x Source #

GLT as row transformations.

Constructors

RowTrafo (GLT x)

Instances

Instances details

Oriented x => Show (RowTrafo x) Source #

Instance details

Defined in OAlg.Entity.Matrix.Transformation

Methods

showsPrec :: Int -> RowTrafo x -> ShowS #

show :: RowTrafo x -> String #

showList :: [RowTrafo x] -> ShowS #

Oriented x => Eq (RowTrafo x) Source #

Instance details

Defined in OAlg.Entity.Matrix.Transformation

Methods

(==) :: RowTrafo x -> RowTrafo x -> Bool #

(/=) :: RowTrafo x -> RowTrafo x -> Bool #

Oriented x => Validable (RowTrafo x) Source #

Instance details

Defined in OAlg.Entity.Matrix.Transformation

Methods

valid :: RowTrafo x -> Statement Source #

Oriented x => Exponential (RowTrafo x) Source #

Instance details

Defined in OAlg.Entity.Matrix.Transformation

Associated Types

type Exponent (RowTrafo x)
Instance details Defined in OAlg.Entity.Matrix.Transformation type Exponent (RowTrafo x) = Z

Methods

(^) :: RowTrafo x -> Exponent (RowTrafo x) -> RowTrafo x Source #

Oriented x => Cayleyan (RowTrafo x) Source #

Instance details

Defined in OAlg.Entity.Matrix.Transformation

Oriented x => Invertible (RowTrafo x) Source #

Instance details

Defined in OAlg.Entity.Matrix.Transformation

Methods

tryToInvert :: RowTrafo x -> Solver (RowTrafo x) Source #

invert :: RowTrafo x -> RowTrafo x Source #

isInvertible :: RowTrafo x -> Bool Source #

zpower :: RowTrafo x -> Z -> RowTrafo x Source #

Oriented x => Multiplicative (RowTrafo x) Source #

Instance details

Defined in OAlg.Entity.Matrix.Transformation

Methods

one :: Point (RowTrafo x) -> RowTrafo x Source #

(*) :: RowTrafo x -> RowTrafo x -> RowTrafo x Source #

npower :: RowTrafo x -> N -> RowTrafo x Source #

Oriented x => Oriented (RowTrafo x) Source #

Instance details

Defined in OAlg.Entity.Matrix.Transformation

Methods

orientation :: RowTrafo x -> Orientation (Point (RowTrafo x)) Source #

start :: RowTrafo x -> Point (RowTrafo x) Source #

end :: RowTrafo x -> Point (RowTrafo x) Source #

Oriented x => EqPoint (RowTrafo x) Source #

Instance details

Defined in OAlg.Entity.Matrix.Transformation

Oriented x => ShowPoint (RowTrafo x) Source #

Instance details

Defined in OAlg.Entity.Matrix.Transformation

Oriented x => TypeablePoint (RowTrafo x) Source #

Instance details

Defined in OAlg.Entity.Matrix.Transformation

Oriented x => ValidablePoint (RowTrafo x) Source #

Instance details

Defined in OAlg.Entity.Matrix.Transformation

Oriented x => Opl (RowTrafo x) (Matrix x) Source #

Instance details

Defined in OAlg.Entity.Matrix.Transformation

Methods

(*>) :: RowTrafo x -> Matrix x -> Matrix x Source #

Distributive x => OrientedOpl (RowTrafo x) (Matrix x) Source #

Instance details

Defined in OAlg.Entity.Matrix.Transformation

type Exponent (RowTrafo x) Source #

Instance details

Defined in OAlg.Entity.Matrix.Transformation

type Exponent (RowTrafo x) = Z

type Point (RowTrafo x) Source #

Instance details

Defined in OAlg.Entity.Matrix.Transformation

type Point (RowTrafo x) = Dim x (Point x)

crTrafoRows :: Transformation x -> Col N (Row N x) -> Col N (Row N x) Source #

applying a transformation as a row transformation on a column of rows.

Col Trafo

newtype ColTrafo x Source #

GLT as a column transformation.

Constructors

ColTrafo (GLT x)

Instances

Instances details

Oriented x => Show (ColTrafo x) Source #

Instance details

Defined in OAlg.Entity.Matrix.Transformation

Methods

showsPrec :: Int -> ColTrafo x -> ShowS #

show :: ColTrafo x -> String #

showList :: [ColTrafo x] -> ShowS #

Oriented x => Eq (ColTrafo x) Source #

Instance details

Defined in OAlg.Entity.Matrix.Transformation

Methods

(==) :: ColTrafo x -> ColTrafo x -> Bool #

(/=) :: ColTrafo x -> ColTrafo x -> Bool #

Oriented x => Validable (ColTrafo x) Source #

Instance details

Defined in OAlg.Entity.Matrix.Transformation

Methods

valid :: ColTrafo x -> Statement Source #

Oriented x => Exponential (ColTrafo x) Source #

Instance details

Defined in OAlg.Entity.Matrix.Transformation

Associated Types

type Exponent (ColTrafo x)
Instance details Defined in OAlg.Entity.Matrix.Transformation type Exponent (ColTrafo x) = Z

Methods

(^) :: ColTrafo x -> Exponent (ColTrafo x) -> ColTrafo x Source #

Oriented x => Cayleyan (ColTrafo x) Source #

Instance details

Defined in OAlg.Entity.Matrix.Transformation

Oriented x => Invertible (ColTrafo x) Source #

Instance details

Defined in OAlg.Entity.Matrix.Transformation

Methods

tryToInvert :: ColTrafo x -> Solver (ColTrafo x) Source #

invert :: ColTrafo x -> ColTrafo x Source #

isInvertible :: ColTrafo x -> Bool Source #

zpower :: ColTrafo x -> Z -> ColTrafo x Source #

Oriented x => Multiplicative (ColTrafo x) Source #

Instance details

Defined in OAlg.Entity.Matrix.Transformation

Methods

one :: Point (ColTrafo x) -> ColTrafo x Source #

(*) :: ColTrafo x -> ColTrafo x -> ColTrafo x Source #

npower :: ColTrafo x -> N -> ColTrafo x Source #

Oriented x => Oriented (ColTrafo x) Source #

Instance details

Defined in OAlg.Entity.Matrix.Transformation

Methods

orientation :: ColTrafo x -> Orientation (Point (ColTrafo x)) Source #

start :: ColTrafo x -> Point (ColTrafo x) Source #

end :: ColTrafo x -> Point (ColTrafo x) Source #

Oriented x => EqPoint (ColTrafo x) Source #

Instance details

Defined in OAlg.Entity.Matrix.Transformation

Oriented x => ShowPoint (ColTrafo x) Source #

Instance details

Defined in OAlg.Entity.Matrix.Transformation

Oriented x => TypeablePoint (ColTrafo x) Source #

Instance details

Defined in OAlg.Entity.Matrix.Transformation

Oriented x => ValidablePoint (ColTrafo x) Source #

Instance details

Defined in OAlg.Entity.Matrix.Transformation

Oriented x => Opr (ColTrafo x) (Matrix x) Source #

Instance details

Defined in OAlg.Entity.Matrix.Transformation

Methods

(<*) :: Matrix x -> ColTrafo x -> Matrix x Source #

Distributive x => OrientedOpr (ColTrafo x) (Matrix x) Source #

Instance details

Defined in OAlg.Entity.Matrix.Transformation

type Exponent (ColTrafo x) Source #

Instance details

Defined in OAlg.Entity.Matrix.Transformation

type Exponent (ColTrafo x) = Z

type Point (ColTrafo x) Source #

Instance details

Defined in OAlg.Entity.Matrix.Transformation

type Point (ColTrafo x) = Dim x (Point x)

crTrafoCols :: Col N (Row N x) -> Transformation x -> Col N (Row N x) Source #

applying a transformation as a column transformation on a column of rows.

Diagonal Form

data DiagonalForm k Source #

the result of transforming a matrix into a diagonal form.

Property Let DiagonalForm ds rt ct be in DiagonalForm k, then holds:

n <= lengthN (start rt) and n <= lengthN (end ct) where n = lengthN ds.
For all d in ds holds: not (isZero d).

Constructors

DiagonalForm [k] (RowTrafo k) (ColTrafo k)

Instances

Instances details

Oriented k => Show (DiagonalForm k) Source #
Instance details Defined in OAlg.Entity.Matrix.Transformation Methods showsPrec :: Int -> DiagonalForm k -> ShowS # show :: DiagonalForm k -> String # showList :: [DiagonalForm k] -> ShowS #
Oriented k => Eq (DiagonalForm k) Source #
Instance details Defined in OAlg.Entity.Matrix.Transformation Methods (==) :: DiagonalForm k -> DiagonalForm k -> Bool # (/=) :: DiagonalForm k -> DiagonalForm k -> Bool #
Distributive k => Validable (DiagonalForm k) Source #
Instance details Defined in OAlg.Entity.Matrix.Transformation Methods valid :: DiagonalForm k -> Statement Source #

dgfMatrix :: Distributive k => DiagonalForm k -> Matrix k Source #

the resulting matrix by applying on the diagonal matrix the inverse of the given transformations.

newtype DiagonalFormStrictPositive k Source #

predicate for diagonal forms with strict positive entries.

Property Let DiagonalFormStrictPositive (DiagonalForm ds _ _) be in DiagonalForm k, then holds: 0 < d for all d in ds.

Constructors

DiagonalFormStrictPositive (DiagonalForm k)

Instances

Instances details

Oriented k => Show (DiagonalFormStrictPositive k) Source #
Instance details Defined in OAlg.Entity.Matrix.Transformation Methods showsPrec :: Int -> DiagonalFormStrictPositive k -> ShowS # show :: DiagonalFormStrictPositive k -> String # showList :: [DiagonalFormStrictPositive k] -> ShowS #
Oriented k => Eq (DiagonalFormStrictPositive k) Source #
Instance details Defined in OAlg.Entity.Matrix.Transformation Methods (==) :: DiagonalFormStrictPositive k -> DiagonalFormStrictPositive k -> Bool # (/=) :: DiagonalFormStrictPositive k -> DiagonalFormStrictPositive k -> Bool #
Number k => Validable (DiagonalFormStrictPositive k) Source #
Instance details Defined in OAlg.Entity.Matrix.Transformation Methods valid :: DiagonalFormStrictPositive k -> Statement Source #