LinearAlgebra: specialize is{hermitian,symmetric} for {Sym,}Tridiagonal

stdlib/LinearAlgebra/src/tridiag.jl

@@ -170,6 +170,9 @@ adjoint(S::SymTridiagonal) = Adjoint(S)
 Base.copy(S::Adjoint{<:Any,<:SymTridiagonal}) = SymTridiagonal(map(x -> copy.(adjoint.(x)), (S.parent.dv, S.parent.ev))...)
 Base.copy(S::Transpose{<:Any,<:SymTridiagonal}) = SymTridiagonal(map(x -> copy.(transpose.(x)), (S.parent.dv, S.parent.ev))...)
 
+ishermitian(S::SymTridiagonal) = isreal(S.dv) && isreal(S.ev)
+issymmetric(S::SymTridiagonal) = true
+
 function diag(M::SymTridiagonal{<:Number}, n::Integer=0)
     # every branch call similar(..., ::Int) to make sure the
     # same vector type is returned independent of n
@@ -608,6 +611,9 @@ transpose(S::Tridiagonal{<:Number}) = Tridiagonal(S.du, S.d, S.dl)
 Base.copy(aS::Adjoint{<:Any,<:Tridiagonal}) = (S = aS.parent; Tridiagonal(map(x -> copy.(adjoint.(x)), (S.du, S.d, S.dl))...))
 Base.copy(tS::Transpose{<:Any,<:Tridiagonal}) = (S = tS.parent; Tridiagonal(map(x -> copy.(transpose.(x)), (S.du, S.d, S.dl))...))
 
+ishermitian(S::Tridiagonal) = isreal(S.d) && S.du == adjoint.(S.dl)
+issymmetric(S::Tridiagonal) = S.du == S.dl
+
 \(A::Adjoint{<:Any,<:Tridiagonal}, B::Adjoint{<:Any,<:StridedVecOrMat}) = copy(A) \ copy(B)
 
 function diag(M::Tridiagonal, n::Integer=0)
@@ -747,7 +753,7 @@ det(A::Tridiagonal) = det_usmani(A.dl, A.d, A.du)
 AbstractMatrix{T}(M::Tridiagonal) where {T} = Tridiagonal{T}(M)
 Tridiagonal{T}(M::SymTridiagonal{T}) where {T} = Tridiagonal(M)
 function SymTridiagonal{T}(M::Tridiagonal) where T
-    if M.dl == M.du
+    if issymmetric(M)
         return SymTridiagonal{T}(convert(AbstractVector{T},M.d), convert(AbstractVector{T},M.dl))
     else
         throw(ArgumentError("Tridiagonal is not symmetric, cannot convert to SymTridiagonal"))

stdlib/LinearAlgebra/test/tridiag.jl

@@ -617,4 +617,32 @@ end
     @test copy(Sc')\b == F'\b
 end
 
+@testset "symmetric and hermitian tridiagonals" begin
+    A = [im 0; 0 -im]
+    @test issymmetric(A)
+    @test !ishermitian(A)
+
+    # real
+    A = SymTridiagonal(randn(5), randn(4))
+    @test issymmetric(A)
+    @test ishermitian(A)
+
+    A = Tridiagonal(A.ev, A.dv, A.ev .+ 1)
+    @test !issymmetric(A)
+    @test !ishermitian(A)
+
+    # complex
+    S = SymTridiagonal(randn(5) .+ 1im, randn(4) .+ 1im)
+    @test issymmetric(S)
+    @test !ishermitian(S)
+
+    S = Tridiagonal(S.ev, S.dv, adjoint.(S.ev))
+    @test !issymmetric(S)
+    @test !ishermitian(S)
+
+    S = Tridiagonal(S.dl, real.(S.d) .+ 0im, S.du)
+    @test !issymmetric(S)
+    @test ishermitian(S)
+end
+
 end # module TestTridiagonal