diff --git a/src/tensors/diagonal.jl b/src/tensors/diagonal.jl
index a13918d03..88d0c3b25 100644
--- a/src/tensors/diagonal.jl
+++ b/src/tensors/diagonal.jl
@@ -37,6 +37,16 @@ storagetype(::Type{<:DiagonalTensorMap{T,S,A}}) where {T,S,A<:DenseVector{T}} =
 
 Construct a `DiagonalTensorMap` with uninitialized data.
 """
+function DiagonalTensorMap{T}(::UndefInitializer, V::TensorMapSpace) where {T}
+    (numin(V) == numout(V) == 1 && domain(V) == codomain(V)) ||
+        throw(ArgumentError("DiagonalTensorMap requires a space with equal domain and codomain and 2 indices"))
+    return DiagonalTensorMap{T}(undef, domain(V))
+end
+function DiagonalTensorMap{T}(::UndefInitializer, V::ProductSpace) where {T}
+    length(V) == 1 ||
+        throw(ArgumentError("DiagonalTensorMap requires `numin(d) == numout(d) == 1`"))
+    return DiagonalTensorMap{T}(undef, only(V))
+end
 function DiagonalTensorMap{T}(::UndefInitializer, V::S) where {T,S<:IndexSpace}
     return DiagonalTensorMap{T,S,Vector{T}}(undef, V)
 end
@@ -265,6 +275,22 @@ function LinearAlgebra.mul!(dC::DiagonalTensorMap,
     return dC
 end
 
+function LinearAlgebra.lmul!(D::DiagonalTensorMap, t::AbstractTensorMap)
+    domain(D) == codomain(t) || throw(SpaceMismatch())
+    for (c, b) in blocks(t)
+        lmul!(block(D, c), b)
+    end
+    return t
+end
+
+function LinearAlgebra.rmul!(t::AbstractTensorMap, D::DiagonalTensorMap)
+    codomain(D) == domain(t) || throw(SpaceMismatch())
+    for (c, b) in blocks(t)
+        rmul!(b, block(D, c))
+    end
+    return t
+end
+
 Base.inv(d::DiagonalTensorMap) = DiagonalTensorMap(inv.(d.data), d.domain)
 function Base.:\(d1::DiagonalTensorMap, d2::DiagonalTensorMap)
     d1.domain == d2.domain || throw(SpaceMismatch())
@@ -339,6 +365,17 @@ function _compute_svddata!(d::DiagonalTensorMap, alg::Union{SVD,SDD})
     return SVDdata, dims
 end
 
+function LinearAlgebra.svdvals(d::DiagonalTensorMap)
+    return SectorDict(c => LinearAlgebra.svdvals(b) for (c, b) in blocks(d))
+end
+function LinearAlgebra.eigvals(d::DiagonalTensorMap)
+    return SectorDict(c => LinearAlgebra.eigvals(b) for (c, b) in blocks(d))
+end
+
+function LinearAlgebra.cond(d::DiagonalTensorMap, p::Real=2)
+    return LinearAlgebra.cond(Diagonal(d.data), p)
+end
+
 # matrix functions
 for f in
     (:exp, :cos, :sin, :tan, :cot, :cosh, :sinh, :tanh, :coth, :atan, :acot, :asinh, :sqrt,
diff --git a/test/diagonal.jl b/test/diagonal.jl
index e4b3f6225..b73597b78 100644
--- a/test/diagonal.jl
+++ b/test/diagonal.jl
@@ -5,8 +5,16 @@ diagspacelist = ((ℂ^4)', ℂ[Z2Irrep](0 => 2, 1 => 3),
 @testset "DiagonalTensor with domain $V" for V in diagspacelist
     @timedtestset "Basic properties and algebra" begin
         for T in (Float32, Float64, ComplexF32, ComplexF64, BigFloat)
+            # constructors
             t = @constinferred DiagonalTensorMap{T}(undef, V)
             t = @constinferred DiagonalTensorMap(rand(T, reduceddim(V)), V)
+            t2 = @constinferred DiagonalTensorMap{T}(undef, space(t))
+            @test space(t2) == space(t)
+            @test_throws ArgumentError DiagonalTensorMap{T}(undef, V^2 ← V)
+            t2 = @constinferred DiagonalTensorMap{T}(undef, domain(t))
+            @test space(t2) == space(t)
+            @test_throws ArgumentError DiagonalTensorMap{T}(undef, V^2)
+            # properties
             @test @constinferred(hash(t)) == hash(deepcopy(t))
             @test scalartype(t) == T
             @test codomain(t) == ProductSpace(V)
@@ -135,6 +143,16 @@ diagspacelist = ((ℂ^4)', ℂ[Z2Irrep](0 => 2, 1 => 3),
         @test u / t1 ≈ u / TensorMap(t1)
         @test t1 * u' ≈ TensorMap(t1) * u'
         @test t1 \ u' ≈ TensorMap(t1) \ u'
+
+        t3 = rand(Float64, V ← V^2)
+        t4 = rand(ComplexF64, V ← V^2)
+        @test t1 * t3 ≈ lmul!(t1, copy(t3))
+        @test t2 * t4 ≈ lmul!(t2, copy(t4))
+
+        t3 = rand(Float64, V^2 ← V)
+        t4 = rand(ComplexF64, V^2 ← V)
+        @test t3 * t1 ≈ rmul!(copy(t3), t1)
+        @test t4 * t2 ≈ rmul!(copy(t4), t2)
     end
     @timedtestset "Tensor contraction" begin
         d = DiagonalTensorMap(rand(ComplexF64, reduceddim(V)), V)
@@ -175,6 +193,12 @@ diagspacelist = ((ℂ^4)', ℂ[Z2Irrep](0 => 2, 1 => 3),
                 VdV2 = V2' * V2
                 @test VdV2 ≈ one(VdV2)
                 @test t2 * V2 ≈ V2 * D2
+
+                @test rank(D) ≈ rank(t)
+                @test cond(D) ≈ cond(t)
+                @test all(((s, t),) -> isapprox(s, t),
+                          zip(values(LinearAlgebra.eigvals(D)),
+                              values(LinearAlgebra.eigvals(t))))
             end
             @testset "leftorth with $alg" for alg in (TensorKit.QR(), TensorKit.QL())
                 Q, R = @constinferred leftorth(t; alg=alg)
@@ -201,6 +225,12 @@ diagspacelist = ((ℂ^4)', ℂ[Z2Irrep](0 => 2, 1 => 3),
                 VdV = Vᴴ * Vᴴ'
                 @test VdV ≈ one(VdV)
                 @test U * S * Vᴴ ≈ t
+
+                @test rank(S) ≈ rank(t)
+                @test cond(S) ≈ cond(t)
+                @test all(((s, t),) -> isapprox(s, t),
+                          zip(values(LinearAlgebra.svdvals(S)),
+                              values(LinearAlgebra.svdvals(t))))
             end
         end
     end