oalg-base-1.1.4.0: Algebraic structures on oriented entities and limits as a tool kit to solve algebraic problems.
Copyright(c) Erich Gut
LicenseBSD3
Maintainerzerich.gut@gmail.com
Safe HaskellSafe-Inferred
LanguageHaskell2010

OAlg.Entity.Matrix.Transformation

Description

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

Synopsis

Row Trafo

newtype RowTrafo a Source #

GLT as row transformations.

Constructors

RowTrafo (GLT a) 

Instances

Instances details
Oriented a => Show (RowTrafo a) Source # 
Instance details

Defined in OAlg.Entity.Matrix.Transformation

Methods

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

show :: RowTrafo a -> String #

showList :: [RowTrafo a] -> ShowS #

Oriented a => Eq (RowTrafo a) Source # 
Instance details

Defined in OAlg.Entity.Matrix.Transformation

Methods

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

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

Oriented a => Validable (RowTrafo a) Source # 
Instance details

Defined in OAlg.Entity.Matrix.Transformation

Methods

valid :: RowTrafo a -> Statement Source #

Oriented a => Entity (RowTrafo a) Source # 
Instance details

Defined in OAlg.Entity.Matrix.Transformation

Oriented a => Exponential (RowTrafo a) Source # 
Instance details

Defined in OAlg.Entity.Matrix.Transformation

Associated Types

type Exponent (RowTrafo a) Source #

Methods

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

Oriented a => Cayleyan (RowTrafo a) Source # 
Instance details

Defined in OAlg.Entity.Matrix.Transformation

Oriented a => Invertible (RowTrafo a) Source # 
Instance details

Defined in OAlg.Entity.Matrix.Transformation

Methods

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

invert :: RowTrafo a -> RowTrafo a Source #

isInvertible :: RowTrafo a -> Bool Source #

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

Oriented a => Multiplicative (RowTrafo a) Source # 
Instance details

Defined in OAlg.Entity.Matrix.Transformation

Methods

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

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

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

Oriented a => Oriented (RowTrafo a) Source # 
Instance details

Defined in OAlg.Entity.Matrix.Transformation

Associated Types

type Point (RowTrafo a) Source #

Methods

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

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

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

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 a) Source # 
Instance details

Defined in OAlg.Entity.Matrix.Transformation

type Exponent (RowTrafo a) = Z
type Point (RowTrafo a) Source # 
Instance details

Defined in OAlg.Entity.Matrix.Transformation

type Point (RowTrafo a) = Dim a (Point a)

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 => Entity (ColTrafo x) Source # 
Instance details

Defined in OAlg.Entity.Matrix.Transformation

Oriented x => Exponential (ColTrafo x) Source # 
Instance details

Defined in OAlg.Entity.Matrix.Transformation

Associated Types

type Exponent (ColTrafo x) Source #

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

Associated Types

type Point (ColTrafo x) Source #

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 => 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:

  1. n <= lengthN (start rt) and n <= lengthN (end ct) where n = lengthN ds.
  2. 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 #

Distributive k => Entity (DiagonalForm k) Source # 
Instance details

Defined in OAlg.Entity.Matrix.Transformation

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 #

Number k => Entity (DiagonalFormStrictPositive k) Source # 
Instance details

Defined in OAlg.Entity.Matrix.Transformation