Complex element type with `SymWoodbury`

[KlausC]

Assuming A and C are symmetric (real or complex) matrices.
In the real case, SymWoodbury(A, U, C) represents A + U * C * U' , because U' == transpose(U).

1. If U is complex, it is unclear, if A + U * C * U' or A + U * C * transpose(U) is meant; the former matrix is not symmetric.

2. Wouldn't it be better to stick to the latter definition for the symmetric case and introduce HermWoodbury(A, U, C), and check if Aand C are hermitian?

3. The checks for symmetry of A and C are not relevant for the validity of the formula, so why are they performed?

4. If a complex A is not symmetric but hermitian, the checks in SymWoodbury can be easily fooled by providing bunchkaufman(A) or cholesky(A) as the first argument.

5. I am not sure, if the generalization using pinv is valid in the symmetric but not real case.

From this confusion result the following errors / inconsistencies:

julia> A = Symmetric(rand(2,2) + rand(2,2)*im);

julia> U = rand(2) + rand(2)*im;

julia> C = [1+im;;]

julia> W = SymWoodbury(A, U, C);

julia> Matrix(W) - (A + U * C * U') # that was expected to be zero
2×2 Matrix{ComplexF64}:
       0.0+0.0im       0.246162-0.246162im
 -0.246162+0.246162im       0.0+0.0im

julia> Matrix(W) - (A + U * C * transpose(U)) # that is another candidate, but wasn't either
2×2 Matrix{ComplexF64}:
 0.418788+0.374783im  1.17165+0.582077im
  1.17165+0.582077im  3.13325+0.742344im

julia> inv(W) # bug?
ERROR: MethodError: no method matching -(::Matrix{ComplexF64}, ::ComplexF64)
For element-wise subtraction, use broadcasting with dot syntax: array .- scalar
...

julia> W \ I # that should be the same as inv(W)
2×2 Matrix{ComplexF64}:
 -0.596091-2.90368im  -0.102938+1.71074im
  0.703646+1.45617im  0.0840259-1.23677im

julia> (A + U *C * U') \ I
2×2 Matrix{ComplexF64}:
 -0.596091-2.90368im  -0.102938+1.71074im
  0.703646+1.45617im  0.0840259-1.23677im

julia> (A + U *C * U') \ I - W \ I # surprise! while the matrices differ, their inverses are equal
2×2 Matrix{ComplexF64}:
 -8.88178e-16+1.33227e-15im   5.55112e-16-8.88178e-16im
  7.77156e-16-6.66134e-16im  -3.88578e-16-2.22045e-16im