Cleanup and generalize functions of Hermitian matrices (#1340 encore)

[araujoms]

No description provided.

[araujoms]

The test failure on Windows is unrelated, the method ambiguities themselves are fixed, as the tests on Linux and Mac show.

[codecov]

Codecov Report

All modified and coverable lines are covered by tests ✅

Project coverage is 93.74%. Comparing base (3e4d569) to head (c970265).
Report is 1 commits behind head on master.

Additional details and impacted files

@@            Coverage Diff             @@
##           master    #1358      +/-   ##
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+ Coverage   93.12%   93.74%   +0.62%     
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  Files          34       34              
  Lines       16705    15724     -981     
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- Hits        15557    14741     -816     
+ Misses       1148      983     -165     

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[araujoms]

I've been thinking about how to test matrix functions on Hermitian{<:Quaternion}, to make sure the capability doesn't get accidentally lost in future refactorings. It would quite overkill to include a spectral eigensolver for quaternions in the test files, so perhaps one could instead mock one with something like

function eigen(A::Hermitian{<:Quaternion})
    U = copy(Matrix(A))
    d = checksquare(A)
    λ = randn(real(eltype(A)), d)
    return Eigen(λ, U)
end

Then one could simply test whether the functions don't throw when called on Hermitian{<:Quaternion}, trusting that correctness would follow from the real and complex cases.

What do you think? Perhaps the entire thing is a waste of time and I should just leave it be.

[dkarrasch]

Hm, perhaps you could design a spectral decomposition right away? Do we have some generic qr, that would give an orthogonal U, and then you specify the real eigenvalues as you like (positive, non-negative, any).

[araujoms]

It's easy to produce an orthogonal U, but I don't see how would that help.

[dkarrasch]

I thought you wanted to test for quaternion matrices, and for that purpose you needed a spectral decomposition. If you start with one, you could multiply out A = U'D*U and use that one to run the functions on? And even compare the results with a manual U' * f.(D) *U? Maybe I misunderstood.

[araujoms]

What I want to test is e.g. that exp(A) works when A is a Hermitian quaternionic matrix. The current implementation of exp will work whenever eigen supports such A (this happens with GenericLinearAlgebra). So in order to test that I don't simply need a spectral decomposition of A, I need eigen to give it to me.