diff --git a/stdlib/LinearAlgebra/src/schur.jl b/stdlib/LinearAlgebra/src/schur.jl
index 9ea03b0bf12a34..d32ee8aeff5044 100644
--- a/stdlib/LinearAlgebra/src/schur.jl
+++ b/stdlib/LinearAlgebra/src/schur.jl
@@ -142,11 +142,47 @@ true
 schur(A::StridedMatrix{<:BlasFloat}) = schur!(copy(A))
 schur(A::StridedMatrix{T}) where T = schur!(copy_oftype(A, eigtype(T)))
 
-schur(A::Symmetric) = schur(copyto!(similar(parent(A)), A))
-schur(A::Hermitian) = schur(copyto!(similar(parent(A)), A))
-schur(A::UpperTriangular) = schur(copyto!(similar(parent(A)), A))
-schur(A::LowerTriangular) = schur(copyto!(similar(parent(A)), A))
-schur(A::Tridiagonal) = schur(Matrix(A))
+schur(A::AbstractMatrix{T}) where {T} = schur!(copyto!(Matrix{eigtype(T)}(undef, size(A)...), A))
+function schur(A::RealHermSymComplexHerm)
+    F = eigen(A; sortby=nothing)
+    return Schur(typeof(F.vectors)(Diagonal(F.values)), F.vectors, F.values)
+end
+function schur(A::Union{UnitUpperTriangular{T},UpperTriangular{T}}) where {T}
+    t = eigtype(T)
+    Z = Matrix{t}(undef, size(A)...)
+    copyto!(Z, A)
+    return Schur(Z, Matrix{t}(I, size(A)), convert(Vector{t}, diag(A)))
+end
+function schur(A::Union{UnitLowerTriangular{T},LowerTriangular{T}}) where {T}
+    t = eigtype(T)
+    # double flip the matrix A
+    Z = Matrix{t}(undef, size(A)...)
+    copyto!(Z, A)
+    reverse!(reshape(Z, :))
+    # construct "reverse" identity
+    n = size(A, 1)
+    J = zeros(t, n, n)
+    for i in axes(J, 2)
+       J[n+1-i, i] = oneunit(t)
+    end
+    return Schur(Z, J, convert(Vector{t}, diag(A)))
+end
+function schur(A::Bidiagonal{T}) where {T}
+    t = eigtype(T)
+    if A.uplo == 'U'
+        return Schur(Matrix{t}(A), Matrix{t}(I, size(A)), Vector{t}(A.dv))
+    else # A.uplo == 'L'
+        # construct "reverse" identity
+        n = size(A, 1)
+        J = zeros(t, n, n)
+        for i in axes(J, 2)
+            J[n+1-i, i] = oneunit(t)
+        end
+        dv = reverse!(Vector{t}(A.dv))
+        ev = reverse!(Vector{t}(A.ev))
+        return Schur(Matrix{t}(Bidiagonal(dv, ev, 'U')), J, dv)
+    end
+end
 
 function getproperty(F::Schur, d::Symbol)
     if d === :Schur
diff --git a/stdlib/LinearAlgebra/test/schur.jl b/stdlib/LinearAlgebra/test/schur.jl
index b660a0700ef952..feb0ef8513b897 100644
--- a/stdlib/LinearAlgebra/test/schur.jl
+++ b/stdlib/LinearAlgebra/test/schur.jl
@@ -37,14 +37,22 @@ aimg  = randn(n,n)/2
 
         sch, vecs, vals = schur(UpperTriangular(triu(a)))
         @test vecs*sch*vecs' ≈ triu(a)
+        sch, vecs, vals = schur(UnitUpperTriangular(triu(a)))
+        @test vecs*sch*vecs' ≈ UnitUpperTriangular(triu(a))
         sch, vecs, vals = schur(LowerTriangular(tril(a)))
         @test vecs*sch*vecs' ≈ tril(a)
+        sch, vecs, vals = schur(UnitLowerTriangular(tril(a)))
+        @test vecs*sch*vecs' ≈ UnitLowerTriangular(tril(a))
         sch, vecs, vals = schur(Hermitian(asym))
         @test vecs*sch*vecs' ≈ asym
         sch, vecs, vals = schur(Symmetric(a + transpose(a)))
         @test vecs*sch*vecs' ≈ a + transpose(a)
         sch, vecs, vals = schur(Tridiagonal(a + transpose(a)))
         @test vecs*sch*vecs' ≈ Tridiagonal(a + transpose(a))
+        sch, vecs, vals = schur(Bidiagonal(a, :U))
+        @test vecs*sch*vecs' ≈ Bidiagonal(a, :U)
+        sch, vecs, vals = schur(Bidiagonal(a, :L))
+        @test vecs*sch*vecs' ≈ Bidiagonal(a, :L)
 
         tstring = sprint((t, s) -> show(t, "text/plain", s), f.T)
         zstring = sprint((t, s) -> show(t, "text/plain", s), f.Z)