Potential error in eigendecomposition with Measurements.jl

[BambOoxX]

Now that #163 has a solution, I've tried to test numeric values for the code.

In addition, I have implemented a custom eigen decomposition under uncertainty using the developments in https://arxiv.org/pdf/2002.05001. Note however that I'm in no way an expert on this theory, I'm just trying to cross-check things.

One thing that is surprising to me is a special case with Hermitian matrices. The paper states that the projection of the uncertainty of the eigenmodes onto their mean values should produce a skew-Hermitian matrix (discussion just below eq (10) in the paper), what I translate as

# Assuming #163 works
A = Symmetric(rand(10,10) .± 0.05.*rand(10,10))
l,p = eigen(A)
Measurements.value.(p)'*Measurements.uncertainty.(p) # Should be skew-symmetric but returns symmetric with non-zero diagonal

My implementation based on the paper provides the expected behavior, but I have some doubts in my own capability to beat specialists on the matter.
Could someone provide some insight into this ? I opened the issue here as it seems more likely to be a linear algebra issue than a Measurements issue but who knows.

The basic verification that (A-l_i*I)*p_i = 0 in Measurements confuses me even more, as it reduces uncertainties to zero in the residuals while my implementation does not.
Given that even 0. ± 0.1 - (0. ± 0.1) does not return a zero uncertainty, something feels fishy, but I am not sure of anything anymore.

[araujoms]

I'm not sure what you're after, but I'm sure this is not a problem with GenericLinearAlgebra. What you can ask for it is to solve the eigenvalue equation, and this it does deliver:

A = Symmetric(rand(10,10) .± 0.05.*rand(10,10))
l, p = eigen(A)
A*p ≈ p*Diagonal(l)

[BambOoxX]

Well, as stated in my first message, I already verified that. The problem is that it seems that Measurements.value.(p)'*Measurements.uncertainty.(p) does not return a skew-hermitian matrix, while according to the paper cited above, it should. I'm just wondering where this inconsistency comes from.

[araujoms]

What I'm saying is that this inconsistency doesn't come from GenericLinearAlgebra. I don't see any reason why Measurements.value.(p)'*Measurements.uncertainty.(p) should return a skew-Hermitian matrix. If it should then Measurements need some notion of eigendecomposition different from the one implemented here.

[BambOoxX]

I don't see any reason why Measurements.value.(p)'*Measurements.uncertainty.(p) should return a skew-Hermitian matrix.

I don't mean to be rude or ungrateful of your help, but I think I was pretty clear when I said twice that this observation comes from the paper cited above.

I am just trying to understand where this discrepancy between different sources comes from, and hopefully contribute to understanding it, and if it's an issue here or in Measurements.jl, or on my part, correct it wherever necessary.

So far, regarding linear algebra operations over Measurements quantities, I've been redirected here, since there seems to be almost zero specialization of linear algebra operations in Measurements itself. Yet, I asked @giordano for help in the matter in JuliaPhysics/Measurements.jl#189 to have his opinion as well, since as you observed Measurements might

need some notion of eigendecomposition different from the one implemented here.