tbd: deprecate array-reducing isreal

[Sacha0]

Deprecation of array-reducing isreal(A::AbstractArray) in favor of all(isinteger, A) sparked debate in #19925, so I've separated that deprecation (into this pull request) from #19925's uncontroversial isinteger deprecation. isreal(A::AbstractArray): keep or nix? Best!

[ararslan]

Personally I'd be in favor of doing away with the array-reducing isreal method since, at least IMHO, the intent of all(isreal, A) is clearer. But if we do this, we should similarly deprecate the array-reducing methods of other functions such as iszero for consistency.

[tkelman]

unrelated, but shouldn't this be !np_real_eigs instead of ~np_real_eigs ?

[Sacha0]

I would have thought so, but apparently ~(x::Bool) is a synonym for !x (~ being bitwise not)?

[tkelman]

they happen to be equivalent for Bool, but I suspect this was written more out of matlab-habit than intentionally

[Sacha0]

Agreed. Will touch that up if this pull request goes forward. Otherwise will open a touchup PR. Thanks!

[stevengj]

My basic argument here is that isreal is a meaningful algebraic question for an array itself — one views the array as a complex algebraic object in its own right, not just as a container for things that have their own algebras (see e.g. complex structure/complexification of vector spaces). This is why we should define real, conj, etcetera for arrays, and for the same reason it is why we should define isreal.

It's analogous to the fact that we define array * scalar, even though this is technically redundant with array .* scalar from a programming standpoint: it's because we want to view arrays mathematically as forming a vector space. Similarly, I think we also want to be able to view complex arrays as a complexified vector space.

[andyferris]

Is all(isreal, x) as fast as e.g. eltype(x) <: Real when that is satisfied?

@stevengj has a point. In common mathematical parlance, you can have a real matrix and you can have a complex matrix. There isn't, in my view, a great deal of semantic mathematical difference between Complex{Matrix{Float64}} and Matrix{Complex{Float64}} - it's just a matter of practicality.