OAlg.Entity.Matrix.Transformation

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

Row Trafo

newtype RowTrafo a

GLT as row transformations.

Constructors

RowTrafo (GLT a)

Instances

Oriented a => Show (RowTrafo a)

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)

Defined in OAlg.Entity.Matrix.Transformation

Methods

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

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

Oriented a => Validable (RowTrafo a)

Defined in OAlg.Entity.Matrix.Transformation

Methods

valid :: RowTrafo a -> Statement

Oriented a => Entity (RowTrafo a)

Defined in OAlg.Entity.Matrix.Transformation

Oriented a => Exponential (RowTrafo a)

Defined in OAlg.Entity.Matrix.Transformation

Associated Types

type Exponent (RowTrafo a)

Methods

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

Oriented a => Cayleyan (RowTrafo a)

Defined in OAlg.Entity.Matrix.Transformation

Oriented a => Invertible (RowTrafo a)

Defined in OAlg.Entity.Matrix.Transformation

Methods

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

invert :: RowTrafo a -> RowTrafo a

isInvertible :: RowTrafo a -> Bool

zpower :: RowTrafo a -> Z -> RowTrafo a

Oriented a => Multiplicative (RowTrafo a)

Defined in OAlg.Entity.Matrix.Transformation

Methods

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

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

npower :: RowTrafo a -> N -> RowTrafo a

Oriented a => Oriented (RowTrafo a)

Defined in OAlg.Entity.Matrix.Transformation

Associated Types

type Point (RowTrafo a)

Methods

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

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

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

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

Defined in OAlg.Entity.Matrix.Transformation

Methods

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

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

Defined in OAlg.Entity.Matrix.Transformation

type Exponent (RowTrafo a)

Defined in OAlg.Entity.Matrix.Transformation

type Exponent (RowTrafo a) = Z

type Point (RowTrafo a)

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)

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

Col Trafo

newtype ColTrafo x

GLT as a column transformation.

Constructors

ColTrafo (GLT x)

Instances

Oriented x => Show (ColTrafo x)

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)

Defined in OAlg.Entity.Matrix.Transformation

Methods

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

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

Oriented x => Validable (ColTrafo x)

Defined in OAlg.Entity.Matrix.Transformation

Methods

valid :: ColTrafo x -> Statement

Oriented x => Entity (ColTrafo x)

Defined in OAlg.Entity.Matrix.Transformation

Oriented x => Exponential (ColTrafo x)

Defined in OAlg.Entity.Matrix.Transformation

Associated Types

type Exponent (ColTrafo x)

Methods

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

Oriented x => Cayleyan (ColTrafo x)

Defined in OAlg.Entity.Matrix.Transformation

Oriented x => Invertible (ColTrafo x)

Defined in OAlg.Entity.Matrix.Transformation

Methods

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

invert :: ColTrafo x -> ColTrafo x

isInvertible :: ColTrafo x -> Bool

zpower :: ColTrafo x -> Z -> ColTrafo x

Oriented x => Multiplicative (ColTrafo x)

Defined in OAlg.Entity.Matrix.Transformation

Methods

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

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

npower :: ColTrafo x -> N -> ColTrafo x

Oriented x => Oriented (ColTrafo x)

Defined in OAlg.Entity.Matrix.Transformation

Associated Types

type Point (ColTrafo x)

Methods

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

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

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

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

Defined in OAlg.Entity.Matrix.Transformation

Methods

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

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

Defined in OAlg.Entity.Matrix.Transformation

type Exponent (ColTrafo x)

Defined in OAlg.Entity.Matrix.Transformation

type Exponent (ColTrafo x) = Z

type Point (ColTrafo x)

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)

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

Diagonal Form

data DiagonalForm k

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

Oriented k => Show (DiagonalForm k)

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)

Defined in OAlg.Entity.Matrix.Transformation

Methods

(==) :: DiagonalForm k -> DiagonalForm k -> Bool

(/=) :: DiagonalForm k -> DiagonalForm k -> Bool

Distributive k => Validable (DiagonalForm k)

Defined in OAlg.Entity.Matrix.Transformation

Methods

valid :: DiagonalForm k -> Statement

Distributive k => Entity (DiagonalForm k)

Defined in OAlg.Entity.Matrix.Transformation

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

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

newtype DiagonalFormStrictPositive k

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

Oriented k => Show (DiagonalFormStrictPositive k)

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)

Defined in OAlg.Entity.Matrix.Transformation

Methods

(==) :: DiagonalFormStrictPositive k -> DiagonalFormStrictPositive k -> Bool

(/=) :: DiagonalFormStrictPositive k -> DiagonalFormStrictPositive k -> Bool

Number k => Validable (DiagonalFormStrictPositive k)

Defined in OAlg.Entity.Matrix.Transformation

Methods

valid :: DiagonalFormStrictPositive k -> Statement

Number k => Entity (DiagonalFormStrictPositive k)

Defined in OAlg.Entity.Matrix.Transformation