diff --git a/Project.toml b/Project.toml
index 8ed52d9..29e0e8a 100644
--- a/Project.toml
+++ b/Project.toml
@@ -1,7 +1,7 @@
 name = "KroneckerArrays"
 uuid = "05d0b138-81bc-4ff7-84be-08becefb1ccc"
 authors = ["ITensor developers <support@itensor.org> and contributors"]
-version = "0.1.4"
+version = "0.1.5"
 
 [deps]
 DerivableInterfaces = "6c5e35bf-e59e-4898-b73c-732dcc4ba65f"
diff --git a/src/KroneckerArrays.jl b/src/KroneckerArrays.jl
index bdedfc7..06fd869 100644
--- a/src/KroneckerArrays.jl
+++ b/src/KroneckerArrays.jl
@@ -582,9 +582,11 @@ using MatrixAlgebraKit:
   eigh_full,
   qr_compact,
   qr_full,
+  left_orth,
   left_polar,
   lq_compact,
   lq_full,
+  right_orth,
   right_polar,
   svd_compact,
   svd_full
@@ -608,12 +610,22 @@ for f in [
   end
 end
 
-for f in (:eig, :eigh, :lq, :qr, :polar, :svd)
-  ff = Symbol("default_", f, "_algorithm")
+for f in [
+  :default_eig_algorithm,
+  :default_eigh_algorithm,
+  :default_lq_algorithm,
+  :default_qr_algorithm,
+  :default_polar_algorithm,
+  :default_svd_algorithm,
+]
   @eval begin
-    function MatrixAlgebraKit.$ff(A::Type{<:KroneckerMatrix}; kwargs...)
+    function MatrixAlgebraKit.$f(
+      A::Type{<:KroneckerMatrix}; kwargs1=(;), kwargs2=(;), kwargs...
+    )
       A1, A2 = argument_types(A)
-      return KroneckerAlgorithm($ff(A1; kwargs...), $ff(A2; kwargs...))
+      return KroneckerAlgorithm(
+        $f(A1; kwargs..., kwargs1...), $f(A2; kwargs..., kwargs2...)
+      )
     end
   end
 end
@@ -631,7 +643,7 @@ function MatrixAlgebraKit.default_algorithm(
   return default_qr_algorithm(A; kwargs...)
 end
 
-for f in (
+for f in [
   :eig_full!,
   :eigh_full!,
   :qr_compact!,
@@ -642,22 +654,24 @@ for f in (
   :right_polar!,
   :svd_compact!,
   :svd_full!,
-)
+]
   @eval begin
     function MatrixAlgebraKit.initialize_output(
       ::typeof($f), a::KroneckerMatrix, alg::KroneckerAlgorithm
     )
       return initialize_output($f, a.a, alg.a) .⊗ initialize_output($f, a.b, alg.b)
     end
-    function MatrixAlgebraKit.$f(a::KroneckerMatrix, F, alg::KroneckerAlgorithm; kwargs...)
-      $f(a.a, Base.Fix2(getfield, :a).(F), alg.a; kwargs...)
-      $f(a.b, Base.Fix2(getfield, :b).(F), alg.b; kwargs...)
+    function MatrixAlgebraKit.$f(
+      a::KroneckerMatrix, F, alg::KroneckerAlgorithm; kwargs1=(;), kwargs2=(;), kwargs...
+    )
+      $f(a.a, Base.Fix2(getfield, :a).(F), alg.a; kwargs..., kwargs1...)
+      $f(a.b, Base.Fix2(getfield, :b).(F), alg.b; kwargs..., kwargs2...)
       return F
     end
   end
 end
 
-for f in (:eig_vals!, :eigh_vals!, :svd_vals!)
+for f in [:eig_vals!, :eigh_vals!, :svd_vals!]
   @eval begin
     function MatrixAlgebraKit.initialize_output(
       ::typeof($f), a::KroneckerMatrix, alg::KroneckerAlgorithm
@@ -672,7 +686,7 @@ for f in (:eig_vals!, :eigh_vals!, :svd_vals!)
   end
 end
 
-for f in (:eig_trunc!, :eigh_trunc!, :svd_trunc!)
+for f in [:eig_trunc!, :eigh_trunc!, :svd_trunc!]
   @eval begin
     function MatrixAlgebraKit.truncate!(
       ::typeof($f),
@@ -684,7 +698,7 @@ for f in (:eig_trunc!, :eigh_trunc!, :svd_trunc!)
   end
 end
 
-for f in (:left_orth!, :right_orth!)
+for f in [:left_orth!, :right_orth!]
   @eval begin
     function MatrixAlgebraKit.initialize_output(::typeof($f), a::KroneckerMatrix)
       return initialize_output($f, a.a) .⊗ initialize_output($f, a.b)
@@ -692,17 +706,155 @@ for f in (:left_orth!, :right_orth!)
   end
 end
 
-for f in (:left_null!, :right_null!)
+for f in [:left_null!, :right_null!]
   @eval begin
     function MatrixAlgebraKit.initialize_output(::typeof($f), a::KroneckerMatrix)
       return initialize_output($f, a.a) ⊗ initialize_output($f, a.b)
     end
-    function MatrixAlgebraKit.$f(a::KroneckerMatrix, F; kwargs...)
-      $f(a.a, F.a; kwargs...)
-      $f(a.b, F.b; kwargs...)
+    function MatrixAlgebraKit.$f(a::KroneckerMatrix, F; kwargs1=(;), kwargs2=(;), kwargs...)
+      $f(a.a, F.a; kwargs..., kwargs1...)
+      $f(a.b, F.b; kwargs..., kwargs2...)
       return F
     end
   end
 end
 
+####################################################################################
+# Special cases for MatrixAlgebraKit factorizations of `Eye(n) ⊗ A` and
+# `A ⊗ Eye(n)` where `A`.
+# TODO: Delete this once https://github.com/QuantumKitHub/MatrixAlgebraKit.jl/pull/34
+# is merged.
+
+using FillArrays: SquareEye
+const SquareEyeKronecker{T,A<:SquareEye{T},B<:AbstractMatrix{T}} = KroneckerMatrix{T,A,B}
+const KroneckerSquareEye{T,A<:AbstractMatrix{T},B<:SquareEye{T}} = KroneckerMatrix{T,A,B}
+const SquareEyeSquareEye{T,A<:SquareEye{T},B<:SquareEye{T}} = KroneckerMatrix{T,A,B}
+
+struct SquareEyeAlgorithm{KWargs<:NamedTuple} <: AbstractAlgorithm
+  kwargs::KWargs
+end
+SquareEyeAlgorithm(; kwargs...) = SquareEyeAlgorithm((; kwargs...))
+
+# Defined to avoid type piracy.
+_copy_input_squareeye(f::F, a) where {F} = copy_input(f, a)
+_copy_input_squareeye(f::F, a::SquareEye) where {F} = a
+
+for f in [
+  :eig_full,
+  :eigh_full,
+  :qr_compact,
+  :qr_full,
+  :left_orth,
+  :left_polar,
+  :lq_compact,
+  :lq_full,
+  :right_orth,
+  :right_polar,
+  :svd_compact,
+  :svd_full,
+]
+  for T in [:SquareEyeKronecker, :KroneckerSquareEye, :SquareEyeSquareEye]
+    @eval begin
+      function MatrixAlgebraKit.copy_input(::typeof($f), a::$T)
+        return _copy_input_squareeye($f, a.a) ⊗ _copy_input_squareeye($f, a.b)
+      end
+    end
+  end
+end
+
+for f in [
+  :default_eig_algorithm,
+  :default_eigh_algorithm,
+  :default_lq_algorithm,
+  :default_qr_algorithm,
+  :default_polar_algorithm,
+  :default_svd_algorithm,
+]
+  f′ = Symbol("_", f, "_squareeye")
+  @eval begin
+    $f′(a; kwargs...) = $f(a; kwargs...)
+    $f′(a::Type{<:SquareEye}; kwargs...) = SquareEyeAlgorithm(; kwargs...)
+  end
+  for T in [:SquareEyeKronecker, :KroneckerSquareEye, :SquareEyeSquareEye]
+    @eval begin
+      function MatrixAlgebraKit.$f(A::Type{<:$T}; kwargs1=(;), kwargs2=(;), kwargs...)
+        A1, A2 = argument_types(A)
+        return KroneckerAlgorithm(
+          $f′(A1; kwargs..., kwargs1...), $f′(A2; kwargs..., kwargs2...)
+        )
+      end
+    end
+  end
+end
+
+# Defined to avoid type piracy.
+_initialize_output_squareeye(f::F, a) where {F} = initialize_output(f, a)
+_initialize_output_squareeye(f::F, a, alg) where {F} = initialize_output(f, a, alg)
+
+for f in [
+  :eig_full!,
+  :eigh_full!,
+  :qr_compact!,
+  :qr_full!,
+  :left_orth!,
+  :left_polar!,
+  :lq_compact!,
+  :lq_full!,
+  :right_orth!,
+  :right_polar!,
+]
+  @eval begin
+    _initialize_output_squareeye(::typeof($f), a::SquareEye) = (a, a)
+    _initialize_output_squareeye(::typeof($f), a::SquareEye, alg) = (a, a)
+  end
+end
+for f in [:svd_compact!, :svd_full!]
+  @eval begin
+    _initialize_output_squareeye(::typeof($f), a::SquareEye) = (a, a, a)
+    _initialize_output_squareeye(::typeof($f), a::SquareEye, alg) = (a, a, a)
+  end
+end
+
+for f in [
+  :eig_full!,
+  :eigh_full!,
+  :qr_compact!,
+  :qr_full!,
+  :left_orth!,
+  :left_polar!,
+  :lq_compact!,
+  :lq_full!,
+  :right_orth!,
+  :right_polar!,
+  :svd_compact!,
+  :svd_full!,
+]
+  f′ = Symbol("_", f, "_squareeye")
+  @eval begin
+    $f′(a, F, alg; kwargs...) = $f(a, F, alg; kwargs...)
+    $f′(a, F, alg::SquareEyeAlgorithm) = F
+  end
+  for T in [:SquareEyeKronecker, :KroneckerSquareEye, :SquareEyeSquareEye]
+    @eval begin
+      function MatrixAlgebraKit.initialize_output(::typeof($f), a::$T)
+        return _initialize_output_squareeye($f, a.a) .⊗
+               _initialize_output_squareeye($f, a.b)
+      end
+      function MatrixAlgebraKit.initialize_output(
+        ::typeof($f), a::$T, alg::KroneckerAlgorithm
+      )
+        return _initialize_output_squareeye($f, a.a, alg.a) .⊗
+               _initialize_output_squareeye($f, a.b, alg.b)
+      end
+      function MatrixAlgebraKit.$f(
+        a::$T, F, alg::KroneckerAlgorithm; kwargs1=(;), kwargs2=(;), kwargs...
+      )
+        $f′(a.a, Base.Fix2(getfield, :a).(F), alg.a; kwargs..., kwargs1...)
+        $f′(a.b, Base.Fix2(getfield, :b).(F), alg.b; kwargs..., kwargs2...)
+        return F
+      end
+    end
+  end
+end
+
 end
diff --git a/test/Project.toml b/test/Project.toml
index 5d58469..aebc8e9 100644
--- a/test/Project.toml
+++ b/test/Project.toml
@@ -7,6 +7,7 @@ MatrixAlgebraKit = "6c742aac-3347-4629-af66-fc926824e5e4"
 SafeTestsets = "1bc83da4-3b8d-516f-aca4-4fe02f6d838f"
 Suppressor = "fd094767-a336-5f1f-9728-57cf17d0bbfb"
 Test = "8dfed614-e22c-5e08-85e1-65c5234f0b40"
+TestExtras = "5ed8adda-3752-4e41-b88a-e8b09835ee3a"
 
 [compat]
 Aqua = "0.8"
@@ -17,3 +18,4 @@ MatrixAlgebraKit = "0.2"
 SafeTestsets = "0.1"
 Suppressor = "0.2"
 Test = "1.10"
+TestExtras = "0.3"
diff --git a/test/test_matrixalgebrakit.jl b/test/test_matrixalgebrakit.jl
index 85822fd..a26508d 100644
--- a/test/test_matrixalgebrakit.jl
+++ b/test/test_matrixalgebrakit.jl
@@ -1,4 +1,5 @@
-using KroneckerArrays: ⊗
+using FillArrays: Eye
+using KroneckerArrays: ⊗, arguments
 using LinearAlgebra: Hermitian, I, diag, hermitianpart, norm
 using MatrixAlgebraKit:
   eig_full,
@@ -22,8 +23,9 @@ using MatrixAlgebraKit:
   svd_trunc,
   svd_vals
 using Test: @test, @test_throws, @testset
+using TestExtras: @constinferred
 
-herm(a) = hermitianpart(a).data
+herm(a) = parent(hermitianpart(a))
 
 @testset "MatrixAlgebraKit" begin
   elt = Float32
@@ -117,3 +119,88 @@ herm(a) = hermitianpart(a).data
   s = svd_vals(a)
   @test s ≈ diag(svd_compact(a)[2])
 end
+
+@testset "MatrixAlgebraKit + Eye" begin
+
+  # TODO:
+  # eig_trunc
+  # eig_vals
+  # eigh_trunc
+  # eigh_vals
+  # left_null
+  # right_null
+  # svd_trunc
+  # svd_vals
+
+  for f in (eig_full, eigh_full)
+    a = Eye(3) ⊗ parent(hermitianpart(randn(3, 3)))
+    d, v = @constinferred f(a)
+    @test a * v ≈ v * d
+    @test arguments(d, 1) isa Eye
+    @test arguments(v, 1) isa Eye
+
+    a = parent(hermitianpart(randn(3, 3))) ⊗ Eye(3)
+    d, v = @constinferred f(a)
+    @test a * v ≈ v * d
+    @test arguments(d, 2) isa Eye
+    @test arguments(v, 2) isa Eye
+
+    a = Eye(3) ⊗ Eye(3)
+    d, v = @constinferred f(a)
+    @test a * v ≈ v * d
+    @test arguments(d, 1) isa Eye
+    @test arguments(d, 2) isa Eye
+    @test arguments(v, 1) isa Eye
+    @test arguments(v, 2) isa Eye
+  end
+
+  for f in (
+    left_orth, left_polar, lq_compact, lq_full, qr_compact, qr_full, right_orth, right_polar
+  )
+    a = Eye(3) ⊗ randn(3, 3)
+    x, y = f(a)
+    @test x * y ≈ a
+    @test arguments(x, 1) isa Eye
+    @test arguments(y, 1) isa Eye
+
+    a = randn(3, 3) ⊗ Eye(3)
+    x, y = f(a)
+    @test x * y ≈ a
+    @test arguments(x, 2) isa Eye
+    @test arguments(y, 2) isa Eye
+
+    a = Eye(3) ⊗ Eye(3)
+    x, y = f(a)
+    @test x * y ≈ a
+    @test arguments(x, 1) isa Eye
+    @test arguments(y, 1) isa Eye
+    @test arguments(x, 2) isa Eye
+    @test arguments(y, 2) isa Eye
+  end
+
+  for f in (svd_compact, svd_full)
+    a = Eye(3) ⊗ randn(3, 3)
+    u, s, v = f(a)
+    @test u * s * v ≈ a
+    @test arguments(u, 1) isa Eye
+    @test arguments(s, 1) isa Eye
+    @test arguments(v, 1) isa Eye
+
+    a = randn(3, 3) ⊗ Eye(3)
+    u, s, v = f(a)
+    @test u * s * v ≈ a
+    @test arguments(u, 2) isa Eye
+    @test arguments(s, 2) isa Eye
+    @test arguments(v, 2) isa Eye
+
+    a = Eye(3) ⊗ Eye(3)
+    u, s, v = f(a)
+    @test u * s * v ≈ a
+    @test arguments(u, 1) isa Eye
+    @test arguments(s, 1) isa Eye
+    @test arguments(v, 1) isa Eye
+    @test arguments(u, 2) isa Eye
+    @test arguments(s, 2) isa Eye
+    @test arguments(v, 2) isa Eye
+  end
+end