diff --git a/stdlib/LinearAlgebra/src/bidiag.jl b/stdlib/LinearAlgebra/src/bidiag.jl
index 28d7b2fe56eb7..0144a575ccfdf 100644
--- a/stdlib/LinearAlgebra/src/bidiag.jl
+++ b/stdlib/LinearAlgebra/src/bidiag.jl
@@ -435,8 +435,76 @@ const BiTri = Union{Bidiagonal,Tridiagonal}
 @inline _mul!(C::AbstractMatrix, A::AbstractMatrix, B::BandedMatrix, alpha::Number, beta::Number) = _mul!(C, A, B, MulAddMul(alpha, beta))
 @inline _mul!(C::AbstractMatrix, A::BandedMatrix, B::BandedMatrix, alpha::Number, beta::Number) = _mul!(C, A, B, MulAddMul(alpha, beta))
 
-lmul!(A::Bidiagonal, B::AbstractVecOrMat) = @inline _mul!(B, A, B, MulAddMul())
-rmul!(B::AbstractMatrix, A::Bidiagonal) = @inline _mul!(B, B, A, MulAddMul())
+# B .= A * B
+function lmul!(A::Bidiagonal, B::AbstractVecOrMat)
+    _muldiag_size_check(A, B)
+    (; dv, ev) = A
+    if A.uplo == 'U'
+        for k in axes(B,2)
+            for i in axes(ev,1)
+                B[i,k] = dv[i] * B[i,k] + ev[i] * B[i+1,k]
+            end
+            B[end,k] = dv[end] * B[end,k]
+        end
+    else
+        for k in axes(B,2)
+            for i in reverse(axes(dv,1)[2:end])
+                B[i,k] = dv[i] * B[i,k] + ev[i-1] * B[i-1,k]
+            end
+            B[1,k] = dv[1] * B[1,k]
+        end
+    end
+    return B
+end
+# B .= D * B
+function lmul!(D::Diagonal, B::Bidiagonal)
+    _muldiag_size_check(D, B)
+    (; dv, ev) = B
+    isL = B.uplo == 'L'
+    dv[1] = D.diag[1] * dv[1]
+    for i in axes(ev,1)
+        ev[i] = D.diag[i + isL] * ev[i]
+        dv[i+1] = D.diag[i+1] * dv[i+1]
+    end
+    return B
+end
+# B .= B * A
+function rmul!(B::AbstractMatrix, A::Bidiagonal)
+    _muldiag_size_check(A, B)
+    (; dv, ev) = A
+    if A.uplo == 'U'
+        for k in reverse(axes(dv,1)[2:end])
+            for i in axes(B,1)
+                B[i,k] = B[i,k] * dv[k] + B[i,k-1] * ev[k-1]
+            end
+        end
+        for i in axes(B,1)
+            B[i,1] *= dv[1]
+        end
+    else
+        for k in axes(ev,1)
+            for i in axes(B,1)
+                B[i,k] = B[i,k] * dv[k] + B[i,k+1] * ev[k]
+            end
+        end
+        for i in axes(B,1)
+            B[i,end] *= dv[end]
+        end
+    end
+    return B
+end
+# B .= B * D
+function rmul!(B::Bidiagonal, D::Diagonal)
+    _muldiag_size_check(B, D)
+    (; dv, ev) = B
+    isU = B.uplo == 'U'
+    dv[1] *= D.diag[1]
+    for i in axes(ev,1)
+        ev[i] *= D.diag[i + isU]
+        dv[i+1] *= D.diag[i+1]
+    end
+    return B
+end
 
 function check_A_mul_B!_sizes(C, A, B)
     mA, nA = size(A)
diff --git a/stdlib/LinearAlgebra/src/diagonal.jl b/stdlib/LinearAlgebra/src/diagonal.jl
index 6ee4f1279b4fd..cd5937e3dcb4f 100644
--- a/stdlib/LinearAlgebra/src/diagonal.jl
+++ b/stdlib/LinearAlgebra/src/diagonal.jl
@@ -310,8 +310,49 @@ function (*)(D::Diagonal, V::AbstractVector)
     return D.diag .* V
 end
 
-rmul!(A::AbstractMatrix, D::Diagonal) = @inline mul!(A, A, D)
-lmul!(D::Diagonal, B::AbstractVecOrMat) = @inline mul!(B, D, B)
+function rmul!(A::AbstractMatrix, D::Diagonal)
+    _muldiag_size_check(A, D)
+    for I in CartesianIndices(A)
+        row, col = Tuple(I)
+        @inbounds A[row, col] *= D.diag[col]
+    end
+    return A
+end
+# T .= T * D
+function rmul!(T::Tridiagonal, D::Diagonal)
+    _muldiag_size_check(T, D)
+    (; dl, d, du) = T
+    d[1] *= D.diag[1]
+    for i in axes(dl,1)
+        dl[i] *= D.diag[i]
+        du[i] *= D.diag[i+1]
+        d[i+1] *= D.diag[i+1]
+    end
+    return T
+end
+
+function lmul!(D::Diagonal, B::AbstractVecOrMat)
+    _muldiag_size_check(D, B)
+    for I in CartesianIndices(B)
+        row = I[1]
+        @inbounds B[I] = D.diag[row] * B[I]
+    end
+    return B
+end
+
+# in-place multiplication with a diagonal
+# T .= D * T
+function lmul!(D::Diagonal, T::Tridiagonal)
+    _muldiag_size_check(D, T)
+    (; dl, d, du) = T
+    d[1] = D.diag[1] * d[1]
+    for i in axes(dl,1)
+        dl[i] = D.diag[i+1] * dl[i]
+        du[i] = D.diag[i] * du[i]
+        d[i+1] = D.diag[i+1] * d[i+1]
+    end
+    return T
+end
 
 function __muldiag!(out, D::Diagonal, B, _add::MulAddMul{ais1,bis0}) where {ais1,bis0}
     require_one_based_indexing(out, B)
diff --git a/stdlib/LinearAlgebra/test/bidiag.jl b/stdlib/LinearAlgebra/test/bidiag.jl
index 3dbb9adf5f562..614b1662f3627 100644
--- a/stdlib/LinearAlgebra/test/bidiag.jl
+++ b/stdlib/LinearAlgebra/test/bidiag.jl
@@ -884,6 +884,41 @@ end
     @test mul!(C1, B, sv, 1, 2) == mul!(C2, B, v, 1 ,2)
 end
 
+@testset "rmul!/lmul! with banded matrices" begin
+    dv, ev = rand(4), rand(3)
+    for A in (Bidiagonal(dv, ev, :U), Bidiagonal(dv, ev, :L))
+        @testset "$(nameof(typeof(B)))" for B in (
+                                Bidiagonal(dv, ev, :U),
+                                Bidiagonal(dv, ev, :L),
+                                Diagonal(dv)
+                        )
+            @test_throws ArgumentError rmul!(B, A)
+            @test_throws ArgumentError lmul!(A, B)
+        end
+        D = Diagonal(dv)
+        @test rmul!(copy(A), D) ≈ A * D
+        @test lmul!(D, copy(A)) ≈ D * A
+    end
+    @testset "non-commutative" begin
+        S32 = SizedArrays.SizedArray{(3,2)}(rand(3,2))
+        S33 = SizedArrays.SizedArray{(3,3)}(rand(3,3))
+        S22 = SizedArrays.SizedArray{(2,2)}(rand(2,2))
+        for uplo in (:L, :U)
+            B = Bidiagonal(fill(S32, 4), fill(S32, 3), uplo)
+            D = Diagonal(fill(S22, size(B,2)))
+            @test rmul!(copy(B), D) ≈ B * D
+            D = Diagonal(fill(S33, size(B,1)))
+            @test lmul!(D, copy(B)) ≈ D * B
+        end
+
+        B = Bidiagonal(fill(S33, 4), fill(S33, 3), :U)
+        D = Diagonal(fill(S32, 4))
+        @test lmul!(B, Array(D)) ≈ B * D
+        B = Bidiagonal(fill(S22, 4), fill(S22, 3), :U)
+        @test rmul!(Array(D), B) ≈ D * B
+    end
+end
+
 @testset "off-band indexing error" begin
     B = Bidiagonal(Vector{BigInt}(undef, 4), Vector{BigInt}(undef,3), :L)
     @test_throws "cannot set entry" B[1,2] = 4
diff --git a/stdlib/LinearAlgebra/test/diagonal.jl b/stdlib/LinearAlgebra/test/diagonal.jl
index 3087e87f63415..8397e97dcdf41 100644
--- a/stdlib/LinearAlgebra/test/diagonal.jl
+++ b/stdlib/LinearAlgebra/test/diagonal.jl
@@ -1288,4 +1288,16 @@ end
     @test yadj == x'
 end
 
+@testset "rmul!/lmul! with banded matrices" begin
+    @testset "$(nameof(typeof(B)))" for B in (
+                            Bidiagonal(rand(4), rand(3), :L),
+                            Tridiagonal(rand(3), rand(4), rand(3))
+                    )
+        BA = Array(B)
+        D = Diagonal(rand(size(B,1)))
+        DA = Array(D)
+        @test rmul!(copy(B), D) ≈ B * D ≈ BA * DA
+        @test lmul!(D, copy(B)) ≈ D * B ≈ DA * BA
+    end
+end
 end # module TestDiagonal
diff --git a/stdlib/LinearAlgebra/test/tridiag.jl b/stdlib/LinearAlgebra/test/tridiag.jl
index 00414fcdc8b56..4f84988cabd50 100644
--- a/stdlib/LinearAlgebra/test/tridiag.jl
+++ b/stdlib/LinearAlgebra/test/tridiag.jl
@@ -833,4 +833,22 @@ end
     @test axes(B) === (ax, ax)
 end
 
+@testset "rmul!/lmul! with banded matrices" begin
+    dl, d, du = rand(3), rand(4), rand(3)
+    A = Tridiagonal(dl, d, du)
+    D = Diagonal(d)
+    @test rmul!(copy(A), D) ≈ A * D
+    @test lmul!(D, copy(A)) ≈ D * A
+
+    @testset "non-commutative" begin
+        S32 = SizedArrays.SizedArray{(3,2)}(rand(3,2))
+        S33 = SizedArrays.SizedArray{(3,3)}(rand(3,3))
+        S22 = SizedArrays.SizedArray{(2,2)}(rand(2,2))
+        T = Tridiagonal(fill(S32,3), fill(S32, 4), fill(S32, 3))
+        D = Diagonal(fill(S22, size(T,2)))
+        @test rmul!(copy(T), D) ≈ T * D
+        D = Diagonal(fill(S33, size(T,1)))
+        @test lmul!(D, copy(T)) ≈ D * T
+    end
+end
 end # module TestTridiagonal
diff --git a/test/testhelpers/SizedArrays.jl b/test/testhelpers/SizedArrays.jl
index 43bc27e630479..bc02fb5cbbd20 100644
--- a/test/testhelpers/SizedArrays.jl
+++ b/test/testhelpers/SizedArrays.jl
@@ -23,6 +23,9 @@ Base.first(::SOneTo) = 1
 Base.last(r::SOneTo) = length(r)
 Base.show(io::IO, r::SOneTo) = print(io, "SOneTo(", length(r), ")")
 
+Broadcast.axistype(a::Base.OneTo, s::SOneTo) = s
+Broadcast.axistype(s::SOneTo, a::Base.OneTo) = s
+
 struct SizedArray{SZ,T,N,A<:AbstractArray} <: AbstractArray{T,N}
     data::A
     function SizedArray{SZ}(data::AbstractArray{T,N}) where {SZ,T,N}
@@ -43,10 +46,25 @@ Base.size(a::SizedArray) = size(typeof(a))
 Base.size(::Type{<:SizedArray{SZ}}) where {SZ} = SZ
 Base.axes(a::SizedArray) = map(SOneTo, size(a))
 Base.getindex(A::SizedArray, i...) = getindex(A.data, i...)
+Base.setindex!(A::SizedArray, v, i...) = setindex!(A.data, v, i...)
 Base.zero(::Type{T}) where T <: SizedArray = SizedArray{size(T)}(zeros(eltype(T), size(T)))
+Base.parent(S::SizedArray) = S.data
 +(S1::SizedArray{SZ}, S2::SizedArray{SZ}) where {SZ} = SizedArray{SZ}(S1.data + S2.data)
 ==(S1::SizedArray{SZ}, S2::SizedArray{SZ}) where {SZ} = S1.data == S2.data
 
+homogenize_shape(t::Tuple) = (_homogenize_shape(first(t)), homogenize_shape(Base.tail(t))...)
+homogenize_shape(::Tuple{}) = ()
+_homogenize_shape(x::Integer) = x
+_homogenize_shape(x::AbstractUnitRange) = length(x)
+const Dims = Union{Integer, Base.OneTo, SOneTo}
+function Base.similar(::Type{A}, shape::Tuple{Dims, Vararg{Dims}}) where {A<:AbstractArray}
+    similar(A, homogenize_shape(shape))
+end
+function Base.similar(::Type{A}, shape::Tuple{SOneTo, Vararg{SOneTo}}) where {A<:AbstractArray}
+    R = similar(A, length.(shape))
+    SizedArray{length.(shape)}(R)
+end
+
 const SizedMatrixLike = Union{SizedMatrix, Transpose{<:Any, <:SizedMatrix}, Adjoint{<:Any, <:SizedMatrix}}
 
 _data(S::SizedArray) = S.data