eigendecomposition of a real symmetric matrix using gsl_eigen_symmv.

 > let (l,v) = eigS $ 'fromLists' [[1,2],[2,1]]
 > l
 3.000 -1.000

 > v
 0.707 -0.707
 0.707  0.707

 > v <> diag l <> trans v
 1.000 2.000
 2.000 1.000

eigendecomposition of a complex hermitian matrix using gsl_eigen_hermv

 > let (l,v) = eigH $ 'fromLists' [[1,2+i],[2-i,3]]

 > l
 4.449 -0.449

 > v
         -0.544          0.839
 (-0.751,0.375) (-0.487,0.243)

 > v <> diag l <> (conjTrans) v
          1.000 (2.000,1.000)
 (2.000,-1.000)         3.000

Singular value decomposition of a real matrix, using gsl_linalg_SV_decomp_mod:

> let (u,s,v) = svdg $ fromLists [[1,2,3],[-4,1,7]]
\
> u
0.310 -0.951
0.951  0.310
\
> s
8.497 2.792
\
> v
-0.411 -0.785
 0.185 -0.570
 0.893 -0.243
\
> u <> diag s <> trans v
 1. 2. 3.
-4. 1. 7.

QR decomposition of a real matrix using gsl_linalg_QR_decomp and gsl_linalg_QR_unpack.

> let (q,r) = qr $ fromLists [[1,3,5,7],[2,0,-2,4]]
\
> q
-0.447 -0.894
-0.894  0.447
\
> r
-2.236 -1.342 -0.447 -6.708
    0. -2.683 -5.367 -4.472
\
> q <> r
1.000 3.000  5.000 7.000
2.000    0. -2.000 4.000

Cholesky decomposition of a symmetric positive definite real matrix using gsl_linalg_cholesky_decomp.

> chol $ (2><2) [1,2,
                   2,9::Double]
(2><2)
 [ 1.0,              0.0
 , 2.0, 2.23606797749979 ]

The LU decomposition of a square matrix. Is based on gsl_linalg_LU_decomp and gsl_linalg_complex_LU_decomp as described in http://www.gnu.org/software/Numeric.GSL/manual/Numeric.GSL-ref_13.html#SEC223.

> let m = fromLists [[1,2,-3],[2+3*i,-7,0],[1,-i,2*i]]
> let (l,u,p,s) = luR m

L is the lower triangular:

> l
          1.            0.  0.
0.154-0.231i            1.  0.
0.154-0.231i  0.624-0.522i  1.

U is the upper triangular:

> u
2.+3.i           -7.            0.
    0.  3.077-1.615i           -3.
    0.            0.  1.873+0.433i

p is a permutation:

> p
[1,0,2]

L * U obtains a permuted version of the original matrix:

> extractRows p m
  2.+3.i   -7.   0.
      1.    2.  -3.
      1.  -1.i  2.i
 
> l <> u
 2.+3.i   -7.   0.
     1.    2.  -3.
     1.  -1.i  2.i

s is the sign of the permutation, required to obtain sign of the determinant:

> s * product (toList $ takeDiag u)
(-18.0) :+ (-16.000000000000004)
> LinearAlgebra.Algorithms.det m
(-18.0) :+ (-16.000000000000004)