Matrix Powers are type-unstable: sometime Real sometimes Complex

[oxinabox]

There are all kinds cases where for some matrix powers the result will be a real matrix,
and for others a complex matrix.
Such as here

julia/stdlib/LinearAlgebra/src/symmetric.jl

Lines 874 to 882 in 70d27c2

	function ^(A::Symmetric{<:Real}, p::Real)
	isinteger(p) && return integerpow(A, p)
	F = eigen(A)
	if all(λ -> λ ≥ 0, F.values)
	return Symmetric((F.vectors * Diagonal((F.values).^p)) * F.vectors')
	else
	return Symmetric((F.vectors * Diagonal((complex(F.values)).^p)) * F.vectors')
	end
	end

For other functions like log it is required that you give suitbale types.
i.e. log(-1) errors, you need to do log(complex(-1))
We should be demanding the same thing from matrix powers.
Give a complex input if you want a complex output.

This was introduced back in #21184

---

Turnout this is really annoying for testing AD.
Making it error would not make it less annoying to test but rather remove the need to test it in the first place.

It breaks FiniteDifferences.jl (JuliaDiff/FiniteDifferences.jl#80) because FIniteDIfferences.jl really can'y handle when the number of inputs needing to be perterbed changes depending on the value of the input.
If it is a complex output there are twice as many things needed to perturbe.

It also really complicates [Zygote's tests](e.g https://github.com/FluxML/Zygote.jl/blob/ac4f1a0727d860b31197a336a02d04b33cb21219/src/lib/array.jl#L607)

[sethaxen]

An example:

julia> using LinearAlgebra, Test

julia> x = [1.0 2.0; 3.0 4.0]
2×2 Array{Float64,2}:
 1.0  2.0
 3.0  4.0

julia> @inferred x^1 # integer power is fine
2×2 Array{Float64,2}:
 1.0  2.0
 3.0  4.0

julia> @inferred x^1.0 # float power is type-unstable
ERROR: return type Array{Float64,2} does not match inferred return type Any
Stacktrace:
 [1] error(::String) at ./error.jl:33
 [2] top-level scope at REPL[23]:1

julia> @inferred x^complex(1.0) # complex power is type-stable
2×2 Array{Complex{Float64},2}:
 1.0-2.35922e-16im  2.0+5.55112e-17im
 3.0-8.32667e-17im  4.0-3.33067e-16im

Another example is eigendecomposition:

julia> @inferred eigen(x)
ERROR: return type Eigen{Float64,Float64,Array{Float64,2},Array{Float64,1}} does not match inferred return type Union{Eigen{Complex{Float64},Complex{Float64},Array{Complex{Float64},2},Array{Complex{Float64},1}}, Eigen{Float64,Float64,Array{Float64,2},Array{Float64,1}}}
Stacktrace:
 [1] error(::String) at ./error.jl:33
 [2] top-level scope at REPL[21]:1