Diagonalizing sparse unitary matrix - julia

Question

I have to gather the eigenvalues of a sparse unitary matrix.
Basically there is just an element different from zero in each
row and column (it's the transfer matrix of some Markovian process).
My question here is how to proceed, what would be the best choice
among all the suite of functions. I have seen that eigs could help,
but I also saw that one has to choose the inital vector.

Answer 1

The following code eventually defines pdeig which returns the eigenvalues of a matrix which is a pdmatrix i.e. a product of a permutation and diagonal matrix, or in other words a matrix like the question describes. Calculating the eigenvectors quickly is also possible (they have an explicit formula):
issquare(m) = all(x->x==size(m,1),size(m))
isunique(v) = v == unique(v)
permmatrix(sigma) =
[i==sigma[j] ? 1.0 : 0.0 for i=1:length(sigma),j=1:length(sigma)]
mat2perm(m) = [findfirst(m[:,i]) for i=1:size(m,1)]
function ispdmatrix(m) # used to verify input matrix form
(r,c,v) = findnz(m)
return issquare(m) && isunique(r) && isunique(c)
end
function pdfact(m::Matrix) # factor into permutation/dilation
ispdmatrix(m) || error("input matrix must be a PD matrix")
n = size(m,1)
p = mat2perm(m)
d = [p[i]>0 ? m[p[i],i] : zero(eltype(m)) for i=1:n]
return (p,d)
end
# return eigenvalues from factored pdmatrix
function pdeig(p::Vector{Int},d::Vector)
n = length(p)
active = trues(n)
eigv = Vector{Complex{eltype(d)}}(0)
for i=1:n
if !active[i]
continue
end
if p[i]>0
j=1
cump = d[i]
k=p[i]
active[i]=false
while active[k] > 0
j+=1
cump *= d[k]
active[k] = false
k=p[k]
end
append!(eigv,[cump^(1.0/j)*exp(2*im*π*m/j) for m=1:j])
else
push!(eigv,0.0 + 0.0im)
end
end
return eigv
end
pdeig(m::Matrix) = pdeig(pdfact(m)...)
n = 4 # testing vector to matrix transformation of permutations
σ=randperm(n)
#assert mat2perm(permmatrix(σ))==σ
For example, the following:
m = [ 0.0 1.0 0.0 ; 2.0 0.0 0.0 ; 0.0 0.0 0.0 ]
pdeig(m)
Outputs:
3-element Array{Complex{Float64},1}:
-1.41421+1.73191e-16im
1.41421-3.46382e-16im
0.0+0.0im
Since these matrices are diagonalizable, the eigenvalues should provide the diagonal matrix (just use diagm on them).
These matrices are very structured, and a proper Julia treatment would define a type for these matrices and then define the various linear algebra functions to dispatch on this type.
In case of errors, just add a comment, and I will try to fix them (or if I happen to see a nice refactoring then I'll edit).
BTW the calculations introduce small numerical errors, these should not be a problem and can be eliminated with proper rounding (so no need to get scared of -1.0 being -1.0+1.234234e-16im)