Eigenvalues not sorted for almost perfectly symmetric matrix

[jeff-regier]

I'm not sure whether this qualifies as a bug, though the behavior seems undesirable. Below A is a matrix that is almost exactly symmetric. eigvals(A) returns the singular values without sorting them. For perfectly symmetric matrices (e.g., B below), the eigenvalues are always sorted. It seems like eigvals(A) should sort them too, since due to numerical issues, even symmetric matrices are often not exactly symmetric.

julia> A = [2.319564160285209 20.613046913136856 -0.5068584177942504
            20.613046913136856 201.80977596527438 -1.2031203195074227
            -0.5068584177942503 -1.203120319507423 11.188307566885028];

julia> eigvals(A)  # these singular values are not sorted
3-element Array{Float64,1}:
 203.926   
   0.198637
  11.1935  

julia> norm(A - A')  # though A is almost perfectly symmetric
2.482534153247273e-16

julia> B = (A + A') / 2.  # but if A is symmetrized
3x3 Array{Float64,2}:
  2.31956    20.613    -0.506858
 20.613     201.81     -1.20312 
 -0.506858   -1.20312  11.1883  

julia> eigvals(B)  # then they are sorted
3-element Array{Float64,1}:
   0.198637
  11.1935  
 203.926   

[stevengj]

If you know you should have a symmetric matrix, you are well-advised to make it perfectly symmetric so that eigvals can use a more efficient algorithm. Maybe the lack of sorting here is a useful clue.

[andreasnoack]

...and remember that you can simply do eigvals(Symmetric(A)) to enforce that a symmetric solver is being used.

[jeff-regier]

I'll use Symmetric in my own program, thanks for suggestion.

Returning eigenvalues out-of-order seems to me like a somewhat opaque way to alert users that a matrix isn't perfectly symmetric. Tiny perturbations of real-valued input shouldn't drastically change the output, if possible.

[KristofferC]

There has been discussion about putting tolerances the symmetry checks but nothing concrete has come out of it.

[jiahao]

Ref: discussion in #182 about approximate symmetry checks in ishermitian and issymmetric

[stevengj]

Closed by JuliaLang/julia#21598.