Is it possible to compute a (strictly) triangular Schur decomposition for a matrix with complex eigenvalues?
With the code below I create such a matrix:
T = rand(2,2)
λ = rand() + rand()*im
D = [real(λ) imag(λ); -imag(λ) real(λ)]
A = T*D/T
schur then returns a quasitriangular decomposition. In my case of a 2x2 matrix with complex eigenvalues:
S,U,Λ = schur(A)
Schur{Float64,Array{Float64,2}}
T factor:
2×2 Array{Float64,2}:
0.921219 -0.0444826
4.41202 0.921219
Z factor:
2×2 Array{Float64,2}:
0.774798 0.632209
-0.632209 0.774798
eigenvalues:
2-element Array{Complex{Float64},1}:
0.9212193623156234 + 0.44301030411320136im
0.9212193623156234 - 0.44301030411320136im
Yes. Do schur(complex(A)).
It would be nice to have a function to get this directly from the quasitriangular decomposition, since that would probably be faster (taking advantage of A being real for most of the computation), but I don’t think that is implemented? (Should be very straightforward to implement since you just need the Schur decompositions of a bunch of 2x2 blocks.)
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Many thanks. Exactly what I needed.
The conversion is implemented as triangularize in the GenericSchur package.
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We might want to move that into the LinearAlgebra stdlib, because some of the matrix-function calculations in LinearAlgebra already need the triangular Schur form and resort to the expensive schur(complex(A)) method to get it IIRC.
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