Ensure inverse of Symmetric{<:,Diagonal} returns same type

[vandenman]

Fixes #1437

Does not add a test for 1438, but let me know if I should.

Feedback is welcome!

[codecov]

Codecov Report

✅ All modified and coverable lines are covered by tests.
✅ Project coverage is 93.90%. Comparing base (98723df) to head (af1fe00).

Additional details and impacted files

@@           Coverage Diff           @@
##           master    #1439   +/-   ##
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  Coverage   93.89%   93.90%           
=======================================
  Files          34       34           
  Lines       15920    15922    +2     
=======================================
+ Hits        14948    14951    +3     
+ Misses        972      971    -1     

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[araujoms]

You do need to add a test for #1438.

You probably want to preserve the type of the number in hermitianpart, to match the behaviour when acting on 1x1 matrices. With something like _hermitianpart(a::T) where {T<:Number} = T(real(a)).

You probably want to do something similar for inv(A::Hermitian{<:Number,<:Diagonal}), it's weird for a complex matrix to suddenly become real.

[stevengj]

Since #1438 is not needed for this PR, it should be a separate PR.

[dkarrasch]

To avoid the issue of returning a real-eltype matrix for a complex-eltype argument, this should perhaps be

Suggested change

	inv(A::Hermitian{<:Number,<:Diagonal}) = Hermitian(inv(real(A.data)), sym_uplo(A.uplo))
	inv(A::Hermitian{T,S}) where {T<:Number, S<:Diagonal} = Hermitian{T,S}(inv(real(A.data)), A.uplo)

?

[vandenman]

I suppose that's possible, but a Hermitian Diagonal matrix is actually always a real Diagonal matrix, so we can deduce from the type alone that the inverse is always a real diagonal matrix. This also holds for quaternion and octonion valued matrices. If there is a number type where becomes a problem, then using real might also be a problem.