'ishermitian' for matrices of variables

[Gavin-Rockwood]

Currently, because of the way ishermitian is implemented in LinearAlgebra and SparseArrays, both functions run into a TypeError: non-boolean (Num) used in boolean context error. Would it be possible to include a method for ishermitian that checks for abstract matrices of symbols?

[ChrisRackauckas]

It should be possible, it would need to use isequals instead... possibly the Base implementation needs to just change to isequals to actually be correct for generic code.

[Gavin-Rockwood]

In my code I have been carrying around a function that that just computes the norm of X-X' and returns isequal(norm, 0). I don't know if there is a better way to do this, but also there are some bits about the multiple dispatch that I don't full understand. If i define this for AbstractMatrix{T} where T<:Number, it will dispatch correctly for normal matrices but not for sparse, even though SparseMatrixCSC{Number} <: AbstractMatrix{T} where T<:Number. I was hoping there might be a more elegant way to do this instead of adding a bunch of methods for different matrix types.

[ChrisRackauckas]

Proving that symbolic expressions are equal or zero is generally undecidable, so this algorithm will have a lot of issues and edge cases... but in theory that can work.

[Gavin-Rockwood]

Well, one one hand this function is needed in order to get symbolics to work with certain packages, which run a ishermitian automatically when setting up structs. So on one hand it does not need to be accurate there. However, if the matrix is purely symbolic, then the norm evaluating to 0 (as in the zero symbol because everything cancels) will be proof enough.

[ChrisRackauckas]

Yes, my point is that it is generally undecidable to prove whether a symbolic expression is equivalent to 0, that's a 1969 theorem. So while you can have an algorithm that checks for this zero, it will necessarily give false negatives

[Gavin-Rockwood]

Sure, but in this instance, if I take M-M'and M is hermitian, then every element will be of the form m-m which is trivially zero. Is there something there I am missing? I'm genuinely curious.

I guess if the elements of M are complicated functions, then I could end up with with some f-g which yes, then I cannot generally determine without either extensive algebra (which still maybe be indeterminable) or numerical evaluation. In either case, being falsely labeled as non-hermitian is not terribly problematic. It is mostly to check if it is obviously hermitian.

[ChrisRackauckas]

You could have sin(x)^2 + cos(x)^2 on [1,2] and 1 on [2,1] in which case you'd have to prove sin(x)^2 + cos(x)^2 - 1 == 0. Of course, that one simplify will get, but you can make the difference be arbitrarily difficult to prove, and it's provably undecidable in general. If you just have polynomial expressions or simple expressions this is fine but if you have some crazy rational mixture of elementary functions it can be quite difficult to see if it's zero.

[Gavin-Rockwood]

Ah! Yeah I was imagining it passing through simplify first, and if the resulting expression is actually 0 then having it return true, and if not then returning zero. In most cases, knowing it is hermitian is the useful part because if it is then you can pass it into some hermitian specific functions. If you don't know then you just treat it as a normal matrix.