MatrixFactorizations.jl

A Julia package to contain non-standard matrix factorizations. At the moment it implements the QL, RQ, and UL factorizations, a combined Cholesky factorization with inverse, and polar decompositions. In the future it may include other factorizations such as the LQ factorization.

QL Factorization

The QL Factorization is analogous to the QR factorization but with a lower-triangular matrix:

julia> using MatrixFactorizations

julia> A = randn(5,5);

julia> ql(A)
MatrixFactorizations.QL{Float64,Array{Float64,2}}
Q factor:
5×5 MatrixFactorizations.QLPackedQ{Float64,Array{Float64,2}}:
  0.155574  -0.112555   -0.686362   -0.701064   0.0233276
  0.363037  -0.0583277  -0.0329428   0.152682   0.916736
 -0.670742   0.294355   -0.547867    0.347157   0.206843
 -0.61094   -0.566041    0.324432   -0.353128   0.276396
  0.144425  -0.759527   -0.349868    0.489877  -0.199681
L factor:
5×5 Array{Float64,2}:
 -1.75734     0.0         0.0        0.0       0.0   
 -0.58336    -2.08515     0.0        0.0       0.0   
  0.0213518   0.0722477  -2.01325    0.0       0.0   
  0.930364   -0.46796    -0.322493  -2.9091    0.0   
  1.08725     0.746217    0.549688  -1.10194  -2.0581

julia> b = randn(5); ql(A) \ b ≈ A \ b
true

Jordan canonical form

Every square matrix has a Jordan canonical form. Although Jordan blocks are unstable with respect to perturbations, the Jordan canonical form of a rational matrix with rational eigenvalues can be found exactly:

julia> A = [0 0 1 7 -1; -5 -6 -6 -35 5; 1 1 -7 7 -1; 0 0 0 -9 0; 2 1 -5 -42 -3];

julia> λ = [-9, -7, -7, -1, -1//1];

julia> V, J = jordan(A, λ)
Jordan{Rational{Int64}, Matrix{Rational{Int64}}, Matrix{Rational{Int64}}}
Generalized eigenvectors:
5×5 Matrix{Rational{Int64}}:
 0  0   0  1   0
 0  1   0  0  -1
 0  1  -1  0   0
 1  0   0  0   0
 7  1  -1  1  -1
Jordan normal form:
5×5 Matrix{Rational{Int64}}:
 -9   0   0   0   0
  0  -7   1   0   0
  0   0  -7   0   0
  0   0   0  -1   1
  0   0   0   0  -1

julia> A == V*J/V
true