Fix tr for block SymTridiagonal (#55371)

This ensures that `tr` for a block `SymTridiagonal` symmetrizes the
diagonal elements.

Author: jishnub

stdlib/LinearAlgebra/src/tridiag.jl

@@ -181,7 +181,7 @@ Base.copy(S::Adjoint{<:Any,<:SymTridiagonal}) = SymTridiagonal(map(x -> copy.(ad
 ishermitian(S::SymTridiagonal) = isreal(S.dv) && isreal(_evview(S))
 issymmetric(S::SymTridiagonal) = true
 
-tr(S::SymTridiagonal) = sum(S.dv)
+tr(S::SymTridiagonal) = sum(symmetric, S.dv)
 
 @noinline function throw_diag_outofboundserror(n, sz)
     sz1, sz2 = sz

stdlib/LinearAlgebra/test/tridiag.jl

@@ -471,7 +471,7 @@ end
 end
 
 @testset "SymTridiagonal/Tridiagonal block matrix" begin
-    M = [1 2; 2 4]
+    M = [1 2; 3 4]
     n = 5
     A = SymTridiagonal(fill(M, n), fill(M, n-1))
     @test @inferred A[1,1] == Symmetric(M)
@@ -485,6 +485,9 @@ end
     @test_throws ArgumentError diag(A, n+1)
     @test_throws ArgumentError diag(A, -n-1)
 
+    @test tr(A) == sum(diag(A))
+    @test issymmetric(tr(A))
+
     A = Tridiagonal(fill(M, n-1), fill(M, n), fill(M, n-1))
     @test @inferred A[1,1] == M
     @test @inferred A[1,2] == M