diff --git a/stdlib/LinearAlgebra/src/LinearAlgebra.jl b/stdlib/LinearAlgebra/src/LinearAlgebra.jl
index ebdb78a0c63c2..bf7f0a4544100 100644
--- a/stdlib/LinearAlgebra/src/LinearAlgebra.jl
+++ b/stdlib/LinearAlgebra/src/LinearAlgebra.jl
@@ -399,7 +399,7 @@ cases where a more specific implementation of `lu!` (or `lu`) is not available.
 
 See also: `copy_oftype`, `copy_similar`
 """
-copy_to_array(A::AbstractArray, ::Type{T}) where {T} = copyto!(Array{T}(undef, size(A)...), A)
+copy_to_array(A::AbstractArray, ::Type{T}) where {T} = copyto!(Array{T}(undef, size(A)), A)
 
 # The three copy functions above return mutable arrays with eltype T.
 # To only ensure a certain eltype, and if a mutable copy is not needed, it is
diff --git a/stdlib/LinearAlgebra/src/bidiag.jl b/stdlib/LinearAlgebra/src/bidiag.jl
index 1a75c6a9287f0..93712e390ff5e 100644
--- a/stdlib/LinearAlgebra/src/bidiag.jl
+++ b/stdlib/LinearAlgebra/src/bidiag.jl
@@ -782,6 +782,41 @@ end
 ldiv!(A::Transpose{<:Any,<:Bidiagonal}, b::AbstractVecOrMat) = ldiv!(copy(A), b)
 ldiv!(A::Adjoint{<:Any,<:Bidiagonal}, b::AbstractVecOrMat) = ldiv!(copy(A), b)
 
+function rdiv!(A::StridedMatrix, B::Bidiagonal)
+    m, n = size(A)
+    if size(B, 1) != n
+        throw(DimensionMismatch("right hand side B needs first dimension of size $n, has size $(size(B,1))"))
+    end
+    if B.uplo == 'L'
+        diagB = B.dv[n]
+        for i = 1:m
+            A[i,n] /= diagB
+        end
+        for j = n-1:-1:1
+            diagB = B.dv[j]
+            offdiagB = B.ev[j]
+            for i = 1:m
+                A[i,j] = (A[i,j] - A[i,j+1]*offdiagB)/diagB
+            end
+        end
+    else
+        diagB = B.dv[1]
+        for i = 1:m
+            A[i,1] /= diagB
+        end
+        for j = 2:n
+            diagB = B.dv[j]
+            offdiagB = B.ev[j-1]
+            for i = 1:m
+                A[i,j] = (A[i,j] - A[i,j-1]*offdiagB)/diagB
+            end
+        end
+    end
+    A
+end
+rdiv!(A::StridedMatrix, B::Adjoint{<:Any,<:Bidiagonal}) = rdiv!(A, copy(B))
+rdiv!(A::StridedMatrix, B::Transpose{<:Any,<:Bidiagonal}) = rdiv!(A, copy(B))
+
 ### Generic promotion methods and fallbacks
 function \(A::Bidiagonal{<:Number}, B::AbstractVecOrMat{<:Number})
     TA, TB = eltype(A), eltype(B)
@@ -806,6 +841,62 @@ end
 
 factorize(A::Bidiagonal) = A
 
+lu!(A::Bidiagonal{T}; check::Bool = true) where {T} =
+    lu!(A, pivot; check = check)
+function lu!(A::Bidiagonal{T}, pivot::NoPivot; check::Bool = true) where {T}
+    if A.uplo == 'U'
+        info = something(findfirst(iszero, A.dv), 0)
+    else
+        info = 0
+        for i in eachindex(A.ev)
+            if iszero(A.ev[i])
+                info = i
+            end
+            A.ev[i] /= A.dv[i]
+        end
+    end
+    check && checknonsingular(info, pivot)
+    return LU{T}(A, 1:length(A.dv), info)
+end
+lu!(A::Bidiagonal{T}, pivot::RowMaximum; check::Bool = true) where {T} =
+    lu!(Tridiagonal{T}(A), pivot; check = check)
+
+lu(A::Bidiagonal{T}, pivot::Union{RowMaximum,NoPivot} = RowMaximum(); check::Bool = true) where {T} =
+    lu!(copy_oftype(A, lutype(T)), pivot; check = check)
+
+function Base.getproperty(F::LU{T,<:Bidiagonal}, d::Symbol) where {T}
+    m, n = size(F)
+    if d === :L
+        L = tril!(getfield(F, :factors)[1:m, 1:min(m,n)])
+        for i = 1:min(m,n); L[i,i] = one(T); end
+        return L
+    elseif d === :U
+        return triu!(getfield(F, :factors)[1:min(m,n), 1:n])
+    elseif d === :p
+        return ipiv2perm(getfield(F, :ipiv), m)
+    elseif d === :P
+        return Matrix{T}(I, m, m)[:,invperm(F.p)]
+    else
+        getfield(F, d)
+    end
+end
+function ldiv!(F::LU{<:Any,<:Bidiagonal}, B::AbstractVecOrMat)
+    Bi = F.factors
+    if Bi.uplo == 'U'
+        return ldiv!(Bi, B)
+    else
+        return ldiv!(Diagonal(Bi.dv), ldiv!(UnitLowerTriangular(Bi), B))
+    end
+end
+function rdiv!(B::AbstractMatrix, F::LU{<:Any,<:Bidiagonal})
+    Bi = F.factors
+    if Bi.uplo == 'U'
+        return rdiv!(B, Bi)
+    else
+        return rdiv!(rdiv!(B, Diagonal(Bi.dv)), UnitLowerTriangular(Bi))
+    end
+end
+
 # Eigensystems
 eigvals(M::Bidiagonal) = M.dv
 function eigvecs(M::Bidiagonal{T}) where T
diff --git a/stdlib/LinearAlgebra/src/diagonal.jl b/stdlib/LinearAlgebra/src/diagonal.jl
index b31360e233a51..96ed69b1fc716 100644
--- a/stdlib/LinearAlgebra/src/diagonal.jl
+++ b/stdlib/LinearAlgebra/src/diagonal.jl
@@ -617,6 +617,34 @@ function getproperty(C::Cholesky{<:Any,<:Diagonal}, d::Symbol)
     end
 end
 
+lu(D::Diagonal{T}, pivot::Union{RowMaximum,NoPivot} = RowMaximum(); check::Bool = true) where {T} =
+    lu!(copy_oftype(D, lutype(T)), pivot, check = check)
+function lu!(D::Diagonal{T}, pivot::Union{RowMaximum,NoPivot} = RowMaximum(); check::Bool = true) where {T}
+    info = something(findfirst(iszero, D.diag), 0)
+    check && checknonsingular(info, pivot)
+    return LU{T}(D, 1:length(D.diag), info)
+end
+
+function Base.getproperty(F::LU{T,<:Diagonal}, d::Symbol) where T
+    m, n = size(F)
+    if d === :L
+        L = tril!(getfield(F, :factors)[1:m, 1:min(m,n)])
+        for i = 1:min(m,n); L[i,i] = one(T); end
+        return L
+    elseif d === :U
+        return triu!(getfield(F, :factors)[1:min(m,n), 1:n])
+    elseif d === :p
+        return ipiv2perm(getfield(F, :ipiv), m)
+    elseif d === :P
+        return Matrix{T}(I, m, m)[:,invperm(F.p)]
+    else
+        getfield(F, d)
+    end
+end
+
+ldiv!(F::LU{<:Any,<:Diagonal}, B::AbstractVecOrMat) = ldiv!(F.factors, B)
+rdiv!(B::AbstractMatrix, F::LU{<:Any,<:Diagonal}) = rdiv!(B, F.factors)
+
 Base._sum(A::Diagonal, ::Colon) = sum(A.diag)
 function Base._sum(A::Diagonal, dims::Integer)
     res = Base.reducedim_initarray(A, dims, zero(eltype(A)))
diff --git a/stdlib/LinearAlgebra/src/lu.jl b/stdlib/LinearAlgebra/src/lu.jl
index 22ed1fc5a49dc..509638d34775e 100644
--- a/stdlib/LinearAlgebra/src/lu.jl
+++ b/stdlib/LinearAlgebra/src/lu.jl
@@ -76,9 +76,6 @@ adjoint(F::LU) = Adjoint(F)
 transpose(F::LU) = Transpose(F)
 
 # StridedMatrix
-lu(A::StridedMatrix, pivot::Union{RowMaximum,NoPivot} = RowMaximum(); check::Bool = true) =
-    lu!(copy_oftype(A, lutype(eltype(A))), pivot; check=check)
-
 lu!(A::StridedMatrix{<:BlasFloat}; check::Bool = true) = lu!(A, RowMaximum(); check=check)
 function lu!(A::StridedMatrix{T}, ::RowMaximum; check::Bool = true) where {T<:BlasFloat}
     lpt = LAPACK.getrf!(A)
@@ -88,10 +85,6 @@ end
 function lu!(A::StridedMatrix{<:BlasFloat}, pivot::NoPivot; check::Bool = true)
     return generic_lufact!(A, pivot; check = check)
 end
-
-lu(A::HermOrSym, pivot::Union{RowMaximum,NoPivot} = RowMaximum(); check::Bool = true) =
-    lu!(copy_oftype(A, lutype(eltype(A))), pivot; check=check)
-
 function lu!(A::HermOrSym, pivot::Union{RowMaximum,NoPivot} = RowMaximum(); check::Bool = true)
     copytri!(A.data, A.uplo, isa(A, Hermitian))
     lu!(A.data, pivot; check = check)
@@ -285,11 +278,12 @@ function lu(A::AbstractMatrix{T}, pivot::Union{RowMaximum,NoPivot} = RowMaximum(
     S = lutype(T)
     lu!(copy_to_array(A, S), pivot; check = check)
 end
+lu(A::HermOrSym, pivot::Union{RowMaximum,NoPivot} = RowMaximum(); check::Bool = true) =
+    lu!(copy_oftype(A, lutype(eltype(A))), pivot; check=check)
 # TODO: remove for Julia v2.0
 @deprecate lu(A::AbstractMatrix, ::Val{true}; check::Bool = true) lu(A, RowMaximum(); check=check)
 @deprecate lu(A::AbstractMatrix, ::Val{false}; check::Bool = true) lu(A, NoPivot(); check=check)
 
-
 lu(S::LU) = S
 function lu(x::Number; check::Bool=true)
     info = x == 0 ? one(BlasInt) : zero(BlasInt)
@@ -755,3 +749,10 @@ AbstractMatrix(F::LU{T,Tridiagonal{T,V}}) where {T,V} = Tridiagonal(F)
 AbstractArray(F::LU{T,Tridiagonal{T,V}}) where {T,V} = AbstractMatrix(F)
 Matrix(F::LU{T,Tridiagonal{T,V}}) where {T,V} = Array(AbstractArray(F))
 Array(F::LU{T,Tridiagonal{T,V}}) where {T,V} = Matrix(F)
+
+# SymTridiagonal
+lu(S::SymTridiagonal{T}, pivot::Union{RowMaximum,NoPivot} = RowMaximum(); check::Bool = true) where {T} =
+    lu!(copy_oftype(Tridiagonal(S), lutype(T)), pivot, check = check)
+function lu!(S::SymTridiagonal{T}, pivot::Union{RowMaximum,NoPivot} = RowMaximum(); check::Bool = true) where {T}
+    return lu!(Tridiagonal(S), pivot, check = check)
+end
diff --git a/stdlib/LinearAlgebra/test/lu.jl b/stdlib/LinearAlgebra/test/lu.jl
index 0dffe7fa1738f..6af3388da38da 100644
--- a/stdlib/LinearAlgebra/test/lu.jl
+++ b/stdlib/LinearAlgebra/test/lu.jl
@@ -399,20 +399,28 @@ end
     end
 end
 
-@testset "lu(A) has a fallback for abstract matrices (#40831)" begin
-    # check that lu works for some structured arrays
-    A0 = rand(5, 5)
-    @test lu(Diagonal(A0)) isa LU
-    @test Matrix(lu(Diagonal(A0))) ≈ Diagonal(A0)
-    @test lu(Bidiagonal(A0, :U)) isa LU
-    @test Matrix(lu(Bidiagonal(A0, :U))) ≈ Bidiagonal(A0, :U)
-
-    # lu(A) copies A and then invokes lu!, make sure that the most efficient
-    # implementation of lu! continues to be used
-    A1 = Tridiagonal(rand(2), rand(3), rand(2))
-    @test lu(A1) isa LU{Float64, Tridiagonal{Float64, Vector{Float64}}}
-    @test lu(A1, RowMaximum()) isa LU{Float64, Tridiagonal{Float64, Vector{Float64}}}
-    @test lu(A1, RowMaximum(); check = false) isa LU{Float64, Tridiagonal{Float64, Vector{Float64}}}
+@testset "lu on *diagonal matrices" begin
+    dl = rand(3)
+    d = rand(4)
+    Bl = Bidiagonal(d, dl, :L)
+    Bu = Bidiagonal(d, dl, :U)
+    Tri = Tridiagonal(dl, d, dl)
+    Sym = SymTridiagonal(d, dl)
+    D = Diagonal(d)
+    b = ones(4)
+    B = rand(4,4)
+    for A in (Bl, Bu, Tri, Sym, D), pivot in (NoPivot(), RowMaximum())
+        @test A\b ≈ lu(A, pivot)\b
+        @test B/A ≈ B/lu(A, pivot)
+        @test B/A ≈ B/Matrix(A)
+        @test Matrix(lu(A, pivot)) ≈ A
+        @test @inferred(lu(A)) isa LU
+        if A isa Union{Tridiagonal, SymTridiagonal}
+            @test lu(A) isa LU{Float64, Tridiagonal{Float64, Vector{Float64}}}
+            @test lu(A, pivot) isa LU{Float64, Tridiagonal{Float64, Vector{Float64}}}
+            @test lu(A, pivot; check = false) isa LU{Float64, Tridiagonal{Float64, Vector{Float64}}}
+        end
+    end
 end
 
 end # module TestLU