Numeric.LAPACK.Matrix.Square

Documentation

type Square sh = Array (Square sh)

size :: Square sh a -> sh

toFull :: (C vert, C horiz) => Square sh a -> Full vert horiz sh sh a

toGeneral :: Square sh a -> General sh sh a

fromGeneral :: Eq sh => General sh sh a -> Square sh a

fromScalar :: Storable a => a -> Square () a

toScalar :: Storable a => Square () a -> a

fromList :: (C sh, Storable a) => sh -> [a] -> Square sh a

autoFromList :: Storable a => [a] -> Square ZeroInt a

transpose :: Square sh a -> Square sh a

adjoint :: (C sh, Floating a) => Square sh a -> Square sh a

conjugate transpose

identity :: (C sh, Floating a) => sh -> Square sh a

identityFrom :: (C sh, Floating a) => Square sh a -> Square sh a

identityFromWidth :: (C height, C width, Floating a) => General height width a -> Square width a

identityFromHeight :: (C height, C width, Floating a) => General height width a -> Square height a

diagonal :: (C sh, Floating a) => Vector sh a -> Square sh a

takeDiagonal :: (C sh, Floating a) => Square sh a -> Vector sh a

trace :: (C sh, Floating a) => Square sh a -> a

multiply :: (C sh, Eq sh, Floating a) => Square sh a -> Square sh a -> Square sh a

square :: (C sh, Floating a) => Square sh a -> Square sh a

power :: (C sh, Floating a) => Integer -> Square sh a -> Square sh a

solve :: (C vert, C horiz, C sh, Eq sh, C nrhs, Floating a) => Square sh a -> Full vert horiz sh nrhs a -> Full vert horiz sh nrhs a

inverse :: (C sh, Floating a) => Square sh a -> Square sh a

determinant :: (C sh, Floating a) => Square sh a -> a

eigenvalues :: (C sh, Floating a) => Square sh a -> Vector sh (ComplexOf a)

schur :: (C sh, Floating a) => Square sh a -> (Square sh a, Square sh a)

If (q,r) = schur a, then a = q <> adjoint q, where q is unitary (orthogonal) and r is a right-upper triangular matrix for complex a and a 1x1-or-2x2-block upper triangular matrix for real a. With takeDiagonal r you get all eigenvalues of a if a is complex and the real parts of the eigenvalues if a is real. Complex conjugated eigenvalues of a real matrix a are encoded as 2x2 blocks along the diagonal.

eigensystem :: (C sh, Floating a) => Square sh a -> (Square sh (ComplexOf a), Vector sh (ComplexOf a), Square sh (ComplexOf a))

(vr,d,vl) = eigensystem a

Counterintuitively, vr contains the right eigenvectors and vl contains the left eigenvectors as columns. The idea is to provide a decomposition of a. If a is diagonalizable, then vr and vl are almost inverse to each other. More precisely, adjoint vl <#> vr is a diagonal matrix. This is because all eigenvectors are normalized to Euclidean norm 1. With the following scaling, the decomposition becomes perfect:

let scal = Array.map recip $ takeDiagonal $ adjoint vl <#> vr
a == vr <#> diagonal d <#> diagonal scal <#> adjoint vl

If a is non-diagonalizable then some columns of vr and vl are left zero and the above property does not hold.

type ComplexOf x = Complex (RealOf x)