Restrict Diagonal sqrt branch to positive diag (#1203)

As noted in #1193,
the `sqrt(::Digonal{<:Real})` method requires the diagonal element to be
positive, so that `sqrt` is defined for the individual elements. We
therefore may restrict the diagonal branch in the dense `sqrt` method to
matrices with positive `diag` if the `eltype` is `Real`.

Fixes #1193

After this,
```julia
julia> A = diagm(0 => [1.0, -1.0])
2×2 Matrix{Float64}:
 1.0   0.0
 0.0  -1.0

julia> sqrt(A)
2×2 Matrix{ComplexF64}:
 1.0+0.0im  0.0+0.0im
 0.0+0.0im  0.0+1.0im
```

(cherry picked from commit 16d9d61)

Author: jishnub

src/dense.jl

@@ -972,7 +972,8 @@ sqrt(::AbstractMatrix)
 function sqrt(A::AbstractMatrix{T}) where {T<:Union{Real,Complex}}
     if checksquare(A) == 0
         return copy(float(A))
-    elseif isdiag(A)
+    elseif isdiag(A) && (T <: Complex || all(x -> x ≥ zero(x), diagview(A)))
+        # Real Diagonal sqrt requires each diagonal element to be positive
         return applydiagonal(sqrt, A)
     elseif ishermitian(A)
         sqrtHermA = sqrt(Hermitian(A))

test/dense.jl

@@ -984,6 +984,12 @@ end
 @testset "sqrt for diagonal" begin
     A = diagm(0 => [1, 2, 3])
     @test sqrt(A)^2 ≈ A
+
+    A = diagm(0 => [1.0, -1.0])
+    @test sqrt(A) == diagm(0 => [1.0, 1.0im])
+    @test sqrt(A)^2 ≈ A
+    B = im*A
+    @test sqrt(B)^2 ≈ B
 end
 
 @testset "issue #40141" begin