lapack-0.2.1: Numerical Linear Algebra using LAPACK

Numeric.LAPACK.Matrix.Square

Synopsis

# Documentation

type Square sh = Array (Square sh)Source

size :: Square sh a -> shSource

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

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

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

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

transpose :: Square sh a -> Square sh aSource

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

conjugate transpose

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

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

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

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

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

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

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

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

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

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

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 aSource

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

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

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

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

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))Source

`(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.