From c53f309326ba28efed3e48752bd118acc8a867b0 Mon Sep 17 00:00:00 2001
From: "copilot-swe-agent[bot]" <198982749+Copilot@users.noreply.github.com>
Date: Thu, 9 Oct 2025 13:12:50 +0000
Subject: [PATCH 1/6] Initial plan


From f622982f78cd63b8bc117d459bddacb5dda8f4db Mon Sep 17 00:00:00 2001
From: "copilot-swe-agent[bot]" <198982749+Copilot@users.noreply.github.com>
Date: Thu, 9 Oct 2025 13:31:34 +0000
Subject: [PATCH 2/6] Rename isisometry to isisometric throughout codebase

Co-authored-by: lkdvos <37111893+lkdvos@users.noreply.github.com>
---
 ext/MatrixAlgebraKitChainRulesCoreExt.jl |  2 +-
 src/MatrixAlgebraKit.jl                  |  2 +-
 src/common/matrixproperties.jl           | 10 ++--
 test/amd/eigh.jl                         |  4 +-
 test/cuda/eigh.jl                        |  4 +-
 test/eigh.jl                             |  4 +-
 test/lq.jl                               | 16 +++----
 test/orthnull.jl                         | 60 ++++++++++++------------
 test/polar.jl                            | 12 ++---
 test/projections.jl                      |  4 +-
 test/qr.jl                               | 18 +++----
 test/schur.jl                            |  2 +-
 test/svd.jl                              | 10 ++--
 13 files changed, 74 insertions(+), 74 deletions(-)

diff --git a/ext/MatrixAlgebraKitChainRulesCoreExt.jl b/ext/MatrixAlgebraKitChainRulesCoreExt.jl
index d8664d80..8e6b930a 100644
--- a/ext/MatrixAlgebraKitChainRulesCoreExt.jl
+++ b/ext/MatrixAlgebraKitChainRulesCoreExt.jl
@@ -14,7 +14,7 @@ MatrixAlgebraKit.iszerotangent(::AbstractZero) = true
 @non_differentiable MatrixAlgebraKit.select_algorithm(args...)
 @non_differentiable MatrixAlgebraKit.initialize_output(args...)
 @non_differentiable MatrixAlgebraKit.check_input(args...)
-@non_differentiable MatrixAlgebraKit.isisometry(args...)
+@non_differentiable MatrixAlgebraKit.isisometric(args...)
 @non_differentiable MatrixAlgebraKit.isunitary(args...)
 
 function ChainRulesCore.rrule(::typeof(copy_input), f, A)
diff --git a/src/MatrixAlgebraKit.jl b/src/MatrixAlgebraKit.jl
index bb6434ab..86c5ef9f 100644
--- a/src/MatrixAlgebraKit.jl
+++ b/src/MatrixAlgebraKit.jl
@@ -9,7 +9,7 @@ using LinearAlgebra: Diagonal, diag, diagind, isdiag
 using LinearAlgebra: UpperTriangular, LowerTriangular
 using LinearAlgebra: BlasFloat, BlasReal, BlasComplex, BlasInt
 
-export isisometry, isunitary, ishermitian, isantihermitian
+export isisometric, isunitary, ishermitian, isantihermitian
 
 export project_hermitian, project_antihermitian, project_isometric
 export project_hermitian!, project_antihermitian!, project_isometric!
diff --git a/src/common/matrixproperties.jl b/src/common/matrixproperties.jl
index 61ff73e9..6f3b6e7b 100644
--- a/src/common/matrixproperties.jl
+++ b/src/common/matrixproperties.jl
@@ -1,5 +1,5 @@
 """
-    isisometry(A; side=:left, isapprox_kwargs...) -> Bool
+    isisometric(A; side=:left, isapprox_kwargs...) -> Bool
 
 Test whether a linear map is an isometry, where the type of isometry is controlled by `kind`:
 
@@ -12,7 +12,7 @@ New specializations should overload [`is_left_isometry`](@ref) and [`is_right_is
 
 See also [`isunitary`](@ref).
 """
-function isisometry(A; side::Symbol = :left, isapprox_kwargs...)
+function isisometric(A; side::Symbol = :left, isapprox_kwargs...)
     side === :left && return is_left_isometry(A; isapprox_kwargs...)
     side === :right && return is_right_isometry(A; isapprox_kwargs...)
 
@@ -25,7 +25,7 @@ end
 Test whether a linear map is unitary, i.e. `A * A' ≈ I ≈ A' * A`.
 The `isapprox_kwargs` are passed on to `isapprox` to control the tolerances.
 
-See also [`isisometry`](@ref).
+See also [`isisometric`](@ref).
 """
 function isunitary(A; isapprox_kwargs...)
     return is_left_isometry(A; isapprox_kwargs...) &&
@@ -42,7 +42,7 @@ end
 Test whether a linear map is a left isometry, i.e. `A' * A ≈ I`.
 The `isapprox_kwargs` can be used to control the tolerances of the equality.
 
-See also [`isisometry`](@ref) and [`is_right_isometry`](@ref).
+See also [`isisometric`](@ref) and [`is_right_isometry`](@ref).
 """ is_left_isometry
 
 function is_left_isometry(A::AbstractMatrix; atol::Real = 0, rtol::Real = defaulttol(A), norm = LinearAlgebra.norm)
@@ -58,7 +58,7 @@ end
 Test whether a linear map is a right isometry, i.e. `A * A' ≈ I`.
 The `isapprox_kwargs` can be used to control the tolerances of the equality.
 
-See also [`isisometry`](@ref) and [`is_left_isometry`](@ref).
+See also [`isisometric`](@ref) and [`is_left_isometry`](@ref).
 """ is_right_isometry
 
 function is_right_isometry(A::AbstractMatrix; atol::Real = 0, rtol::Real = defaulttol(A), norm = LinearAlgebra.norm)
diff --git a/test/amd/eigh.jl b/test/amd/eigh.jl
index 0e2cb5be..d3cd25f4 100644
--- a/test/amd/eigh.jl
+++ b/test/amd/eigh.jl
@@ -48,14 +48,14 @@ end
 
         D1, V1 = @constinferred eigh_trunc(A; alg, trunc=truncrank(r))
         @test length(diagview(D1)) == r
-        @test isisometry(V1)
+        @test isisometric(V1)
         @test A * V1 ≈ V1 * D1
         @test LinearAlgebra.opnorm(A - V1 * D1 * V1') ≈ D₀[r + 1]
 
         trunc = trunctol(; atol=s * D₀[r + 1])
         D2, V2 = @constinferred eigh_trunc(A; alg, trunc)
         @test length(diagview(D2)) == r
-        @test isisometry(V2)
+        @test isisometric(V2)
         @test A * V2 ≈ V2 * D2
 
         # test for same subspace
diff --git a/test/cuda/eigh.jl b/test/cuda/eigh.jl
index eece6be3..0a9be80d 100644
--- a/test/cuda/eigh.jl
+++ b/test/cuda/eigh.jl
@@ -43,14 +43,14 @@ end
 
         D1, V1 = @constinferred eigh_trunc(A; alg, trunc=truncrank(r))
         @test length(diagview(D1)) == r
-        @test isisometry(V1)
+        @test isisometric(V1)
         @test A * V1 ≈ V1 * D1
         @test LinearAlgebra.opnorm(A - V1 * D1 * V1') ≈ D₀[r + 1]
 
         trunc = trunctol(; atol = s * D₀[r + 1])
         D2, V2 = @constinferred eigh_trunc(A; alg, trunc)
         @test length(diagview(D2)) == r
-        @test isisometry(V2)
+        @test isisometric(V2)
         @test A * V2 ≈ V2 * D2
 
         # test for same subspace
diff --git a/test/eigh.jl b/test/eigh.jl
index fa766abb..74f50604 100644
--- a/test/eigh.jl
+++ b/test/eigh.jl
@@ -52,14 +52,14 @@ end
 
         D1, V1 = @constinferred eigh_trunc(A; alg, trunc = truncrank(r))
         @test length(diagview(D1)) == r
-        @test isisometry(V1)
+        @test isisometric(V1)
         @test A * V1 ≈ V1 * D1
         @test LinearAlgebra.opnorm(A - V1 * D1 * V1') ≈ D₀[r + 1]
 
         trunc = trunctol(; atol = s * D₀[r + 1])
         D2, V2 = @constinferred eigh_trunc(A; alg, trunc)
         @test length(diagview(D2)) == r
-        @test isisometry(V2)
+        @test isisometric(V2)
         @test A * V2 ≈ V2 * D2
 
         s = 1 - sqrt(eps(real(T)))
diff --git a/test/lq.jl b/test/lq.jl
index 5a52b601..2c8dfefe 100644
--- a/test/lq.jl
+++ b/test/lq.jl
@@ -17,11 +17,11 @@ eltypes = (Float32, Float64, ComplexF32, ComplexF64)
         @test L isa Matrix{T} && size(L) == (m, minmn)
         @test Q isa Matrix{T} && size(Q) == (minmn, n)
         @test L * Q ≈ A
-        @test isisometry(Q; side = :right)
+        @test isisometric(Q; side = :right)
         Nᴴ = @constinferred lq_null(A)
         @test Nᴴ isa Matrix{T} && size(Nᴴ) == (n - minmn, n)
         @test maximum(abs, A * Nᴴ') < eps(real(T))^(2 / 3)
-        @test isisometry(Nᴴ; side = :right)
+        @test isisometric(Nᴴ; side = :right)
 
         Ac = similar(A)
         L2, Q2 = @constinferred lq_compact!(copy!(Ac, A), (L, Q))
@@ -51,12 +51,12 @@ eltypes = (Float32, Float64, ComplexF32, ComplexF64)
         # unblocked algorithm
         lq_compact!(copy!(Ac, A), (L, Q); blocksize = 1)
         @test L * Q ≈ A
-        @test isisometry(Q; side = :right)
+        @test isisometric(Q; side = :right)
         lq_compact!(copy!(Ac, A), (noL, Q2); blocksize = 1)
         @test Q == Q2
         lq_null!(copy!(Ac, A), Nᴴ; blocksize = 1)
         @test maximum(abs, A * Nᴴ') < eps(real(T))^(2 / 3)
-        @test isisometry(Nᴴ; side = :right)
+        @test isisometric(Nᴴ; side = :right)
         if m <= n
             lq_compact!(copy!(Q2, A), (noL, Q2); blocksize = 1) # in-place Q
             @test Q ≈ Q2
@@ -66,12 +66,12 @@ eltypes = (Float32, Float64, ComplexF32, ComplexF64)
         end
         lq_compact!(copy!(Ac, A), (L, Q); blocksize = 8)
         @test L * Q ≈ A
-        @test isisometry(Q; side = :right)
+        @test isisometric(Q; side = :right)
         lq_compact!(copy!(Ac, A), (noL, Q2); blocksize = 8)
         @test Q == Q2
         lq_null!(copy!(Ac, A), Nᴴ; blocksize = 8)
         @test maximum(abs, A * Nᴴ') < eps(real(T))^(2 / 3)
-        @test isisometry(Nᴴ; side = :right)
+        @test isisometric(Nᴴ; side = :right)
         @test Nᴴ * Nᴴ' ≈ I
 
         qr_alg = LAPACK_HouseholderQR(; blocksize = 1)
@@ -91,7 +91,7 @@ eltypes = (Float32, Float64, ComplexF32, ComplexF64)
         # positive
         lq_compact!(copy!(Ac, A), (L, Q); positive = true)
         @test L * Q ≈ A
-        @test isisometry(Q; side = :right)
+        @test isisometric(Q; side = :right)
         @test all(>=(zero(real(T))), real(diag(L)))
         lq_compact!(copy!(Ac, A), (noL, Q2); positive = true)
         @test Q == Q2
@@ -99,7 +99,7 @@ eltypes = (Float32, Float64, ComplexF32, ComplexF64)
         # positive and blocksize 1
         lq_compact!(copy!(Ac, A), (L, Q); positive = true, blocksize = 1)
         @test L * Q ≈ A
-        @test isisometry(Q; side = :right)
+        @test isisometric(Q; side = :right)
         @test all(>=(zero(real(T))), real(diag(L)))
         lq_compact!(copy!(Ac, A), (noL, Q2); positive = true, blocksize = 1)
         @test Q == Q2
diff --git a/test/orthnull.jl b/test/orthnull.jl
index eac92be3..fc364916 100644
--- a/test/orthnull.jl
+++ b/test/orthnull.jl
@@ -68,9 +68,9 @@ end
         @test C isa Matrix{T} && size(C) == (minmn, n)
         @test N isa Matrix{T} && size(N) == (m, m - minmn)
         @test V * C ≈ A
-        @test isisometry(V)
+        @test isisometric(V)
         @test LinearAlgebra.norm(A' * N) ≈ 0 atol = MatrixAlgebraKit.defaulttol(T)
-        @test isisometry(N)
+        @test isisometric(N)
         @test V * V' + N * N' ≈ I
 
         M = LinearMap(A)
@@ -85,9 +85,9 @@ end
             @test C isa Matrix{T} && size(C) == (minmn, n)
             @test N isa Matrix{T} && size(N) == (m, nullity)
             @test V * C ≈ A
-            @test isisometry(V)
+            @test isisometric(V)
             @test LinearAlgebra.norm(A' * N) ≈ 0 atol = MatrixAlgebraKit.defaulttol(T)
-            @test isisometry(N)
+            @test isisometric(N)
         end
 
         for alg_qr in ((; positive = true), (; positive = false), LAPACK_HouseholderQR())
@@ -97,9 +97,9 @@ end
             @test C isa Matrix{T} && size(C) == (minmn, n)
             @test N isa Matrix{T} && size(N) == (m, m - minmn)
             @test V * C ≈ A
-            @test isisometry(V)
+            @test isisometric(V)
             @test LinearAlgebra.norm(A' * N) ≈ 0 atol = MatrixAlgebraKit.defaulttol(T)
-            @test isisometry(N)
+            @test isisometric(N)
             @test V * V' + N * N' ≈ I
         end
 
@@ -110,9 +110,9 @@ end
         @test C2 === C
         @test N2 === N
         @test V2 * C2 ≈ A
-        @test isisometry(V2)
+        @test isisometric(V2)
         @test LinearAlgebra.norm(A' * N2) ≈ 0 atol = MatrixAlgebraKit.defaulttol(T)
-        @test isisometry(N2)
+        @test isisometric(N2)
         @test V2 * V2' + N2 * N2' ≈ I
 
         atol = eps(real(T))
@@ -122,9 +122,9 @@ end
         @test C2 !== C
         @test N2 !== C
         @test V2 * C2 ≈ A
-        @test isisometry(V2)
+        @test isisometric(V2)
         @test LinearAlgebra.norm(A' * N2) ≈ 0 atol = MatrixAlgebraKit.defaulttol(T)
-        @test isisometry(N2)
+        @test isisometric(N2)
         @test V2 * V2' + N2 * N2' ≈ I
 
         rtol = eps(real(T))
@@ -138,9 +138,9 @@ end
             @test C2 !== C
             @test N2 !== C
             @test V2 * C2 ≈ A
-            @test isisometry(V2)
+            @test isisometric(V2)
             @test LinearAlgebra.norm(A' * N2) ≈ 0 atol = MatrixAlgebraKit.defaulttol(T)
-            @test isisometry(N2)
+            @test isisometric(N2)
             @test V2 * V2' + N2 * N2' ≈ I
         end
 
@@ -150,12 +150,12 @@ end
             @test V2 === V
             @test C2 === C
             @test V2 * C2 ≈ A
-            @test isisometry(V2)
+            @test isisometric(V2)
             if kind != :polar
                 N2 = @constinferred left_null!(copy!(Ac, A), N; kind = kind)
                 @test N2 === N
                 @test LinearAlgebra.norm(A' * N2) ≈ 0 atol = MatrixAlgebraKit.defaulttol(T)
-                @test isisometry(N2)
+                @test isisometric(N2)
                 @test V2 * V2' + N2 * N2' ≈ I
             end
 
@@ -178,9 +178,9 @@ end
                 @test C2 !== C
                 @test N2 !== C
                 @test V2 * C2 ≈ A
-                @test isisometry(V2)
+                @test isisometric(V2)
                 @test LinearAlgebra.norm(A' * N2) ≈ 0 atol = MatrixAlgebraKit.defaulttol(T)
-                @test isisometry(N2)
+                @test isisometric(N2)
                 @test V2 * V2' + N2 * N2' ≈ I
             else
                 @test_throws ArgumentError left_orth!(copy!(Ac, A), (V, C); kind, trunc = (; atol))
@@ -204,9 +204,9 @@ end
         @test Vᴴ isa Matrix{T} && size(Vᴴ) == (minmn, n)
         @test Nᴴ isa Matrix{T} && size(Nᴴ) == (n - minmn, n)
         @test C * Vᴴ ≈ A
-        @test isisometry(Vᴴ; side = :right)
+        @test isisometric(Vᴴ; side = :right)
         @test LinearAlgebra.norm(A * adjoint(Nᴴ)) ≈ 0 atol = MatrixAlgebraKit.defaulttol(T)
-        @test isisometry(Nᴴ; side = :right)
+        @test isisometric(Nᴴ; side = :right)
         @test Vᴴ' * Vᴴ + Nᴴ' * Nᴴ ≈ I
 
         M = LinearMap(A)
@@ -220,9 +220,9 @@ end
         @test Vᴴ2 === Vᴴ
         @test Nᴴ2 === Nᴴ
         @test C2 * Vᴴ2 ≈ A
-        @test isisometry(Vᴴ2; side = :right)
+        @test isisometric(Vᴴ2; side = :right)
         @test LinearAlgebra.norm(A * adjoint(Nᴴ2)) ≈ 0 atol = MatrixAlgebraKit.defaulttol(T)
-        @test isisometry(Nᴴ; side = :right)
+        @test isisometric(Nᴴ; side = :right)
         @test Vᴴ2' * Vᴴ2 + Nᴴ2' * Nᴴ2 ≈ I
 
         atol = eps(real(T))
@@ -232,9 +232,9 @@ end
         @test Vᴴ2 !== Vᴴ
         @test Nᴴ2 !== Nᴴ
         @test C2 * Vᴴ2 ≈ A
-        @test isisometry(Vᴴ2; side = :right)
+        @test isisometric(Vᴴ2; side = :right)
         @test LinearAlgebra.norm(A * adjoint(Nᴴ2)) ≈ 0 atol = MatrixAlgebraKit.defaulttol(T)
-        @test isisometry(Nᴴ; side = :right)
+        @test isisometric(Nᴴ; side = :right)
         @test Vᴴ2' * Vᴴ2 + Nᴴ2' * Nᴴ2 ≈ I
 
         rtol = eps(real(T))
@@ -244,9 +244,9 @@ end
         @test Vᴴ2 !== Vᴴ
         @test Nᴴ2 !== Nᴴ
         @test C2 * Vᴴ2 ≈ A
-        @test isisometry(Vᴴ2; side = :right)
+        @test isisometric(Vᴴ2; side = :right)
         @test LinearAlgebra.norm(A * adjoint(Nᴴ2)) ≈ 0 atol = MatrixAlgebraKit.defaulttol(T)
-        @test isisometry(Nᴴ2; side = :right)
+        @test isisometric(Nᴴ2; side = :right)
         @test Vᴴ2' * Vᴴ2 + Nᴴ2' * Nᴴ2 ≈ I
 
         for kind in (:lq, :polar, :svd)
@@ -255,12 +255,12 @@ end
             @test C2 === C
             @test Vᴴ2 === Vᴴ
             @test C2 * Vᴴ2 ≈ A
-            @test isisometry(Vᴴ2; side = :right)
+            @test isisometric(Vᴴ2; side = :right)
             if kind != :polar
                 Nᴴ2 = @constinferred right_null!(copy!(Ac, A), Nᴴ; kind)
                 @test Nᴴ2 === Nᴴ
                 @test LinearAlgebra.norm(A * adjoint(Nᴴ2)) ≈ 0 atol = MatrixAlgebraKit.defaulttol(T)
-                @test isisometry(Nᴴ2; side = :right)
+                @test isisometric(Nᴴ2; side = :right)
                 @test Vᴴ2' * Vᴴ2 + Nᴴ2' * Nᴴ2 ≈ I
             end
 
@@ -271,9 +271,9 @@ end
                 @test Vᴴ2 !== Vᴴ
                 @test Nᴴ2 !== Nᴴ
                 @test C2 * Vᴴ2 ≈ A
-                @test isisometry(Vᴴ2; side = :right)
+                @test isisometric(Vᴴ2; side = :right)
                 @test LinearAlgebra.norm(A * adjoint(Nᴴ2)) ≈ 0 atol = MatrixAlgebraKit.defaulttol(T)
-                @test isisometry(Nᴴ2; side = :right)
+                @test isisometric(Nᴴ2; side = :right)
                 @test Vᴴ2' * Vᴴ2 + Nᴴ2' * Nᴴ2 ≈ I
 
                 C2, Vᴴ2 = @constinferred right_orth!(copy!(Ac, A), (C, Vᴴ); kind, trunc = (; rtol))
@@ -282,9 +282,9 @@ end
                 @test Vᴴ2 !== Vᴴ
                 @test Nᴴ2 !== Nᴴ
                 @test C2 * Vᴴ2 ≈ A
-                @test isisometry(Vᴴ2; side = :right)
+                @test isisometric(Vᴴ2; side = :right)
                 @test LinearAlgebra.norm(A * adjoint(Nᴴ2)) ≈ 0 atol = MatrixAlgebraKit.defaulttol(T)
-                @test isisometry(Nᴴ2; side = :right)
+                @test isisometric(Nᴴ2; side = :right)
                 @test Vᴴ2' * Vᴴ2 + Nᴴ2' * Nᴴ2 ≈ I
             else
                 @test_throws ArgumentError right_orth!(copy!(Ac, A), (C, Vᴴ); kind, trunc = (; atol))
diff --git a/test/polar.jl b/test/polar.jl
index 9e4ee341..8087149e 100644
--- a/test/polar.jl
+++ b/test/polar.jl
@@ -22,7 +22,7 @@ using LinearAlgebra: LinearAlgebra, I, isposdef
             @test W isa Matrix{T} && size(W) == (m, n)
             @test P isa Matrix{T} && size(P) == (n, n)
             @test W * P ≈ A
-            @test isisometry(W)
+            @test isisometric(W)
             @test isposdef(P)
 
             Ac = similar(A)
@@ -30,14 +30,14 @@ using LinearAlgebra: LinearAlgebra, I, isposdef
             @test W2 === W
             @test P2 === P
             @test W * P ≈ A
-            @test isisometry(W)
+            @test isisometric(W)
             @test isposdef(P)
 
             noP = similar(P, (0, 0))
             W2, P2 = @constinferred left_polar!(copy!(Ac, A), (W, noP), alg)
             @test P2 === noP
             @test W2 === W
-            @test isisometry(W)
+            @test isisometric(W)
             P = W' * A # compute P explicitly to verify W correctness
             @test ishermitian(P; rtol = MatrixAlgebraKit.defaulttol(P))
             @test isposdef(project_hermitian!(P))
@@ -59,7 +59,7 @@ end
             @test Wᴴ isa Matrix{T} && size(Wᴴ) == (m, n)
             @test P isa Matrix{T} && size(P) == (m, m)
             @test P * Wᴴ ≈ A
-            @test isisometry(Wᴴ; side = :right)
+            @test isisometric(Wᴴ; side = :right)
             @test isposdef(P)
 
             Ac = similar(A)
@@ -67,14 +67,14 @@ end
             @test P2 === P
             @test Wᴴ2 === Wᴴ
             @test P * Wᴴ ≈ A
-            @test isisometry(Wᴴ; side = :right)
+            @test isisometric(Wᴴ; side = :right)
             @test isposdef(P)
 
             noP = similar(P, (0, 0))
             P2, Wᴴ2 = @constinferred right_polar!(copy!(Ac, A), (noP, Wᴴ), alg)
             @test P2 === noP
             @test Wᴴ2 === Wᴴ
-            @test isisometry(Wᴴ; side = :right)
+            @test isisometric(Wᴴ; side = :right)
             P = A * Wᴴ' # compute P explicitly to verify W correctness
             @test ishermitian(P; rtol = MatrixAlgebraKit.defaulttol(P))
             @test isposdef(project_hermitian!(P))
diff --git a/test/projections.jl b/test/projections.jl
index 5698a702..684b3bac 100644
--- a/test/projections.jl
+++ b/test/projections.jl
@@ -59,7 +59,7 @@ end
         @testset "algorithm $alg" for alg in algs
             A = randn(rng, T, m, n)
             W = project_isometric(A, alg)
-            @test isisometry(W)
+            @test isisometric(W)
             W2 = project_isometric(W, alg)
             @test W2 ≈ W # stability of the projection
             @test W * (W' * A) ≈ A
@@ -67,7 +67,7 @@ end
             Ac = similar(A)
             W2 = @constinferred project_isometric!(copy!(Ac, A), W, alg)
             @test W2 === W
-            @test isisometry(W)
+            @test isisometric(W)
 
             # test that W is closer to A then any other isometry
             for k in 1:10
diff --git a/test/qr.jl b/test/qr.jl
index 39d4a2ac..826c320b 100644
--- a/test/qr.jl
+++ b/test/qr.jl
@@ -18,9 +18,9 @@ eltypes = (Float32, Float64, ComplexF32, ComplexF64)
         @test Q * R ≈ A
         N = @constinferred qr_null(A)
         @test N isa Matrix{T} && size(N) == (m, m - minmn)
-        @test isisometry(Q)
+        @test isisometric(Q)
         @test maximum(abs, A' * N) < eps(real(T))^(2 / 3)
-        @test isisometry(N)
+        @test isisometric(N)
 
         Ac = similar(A)
         Q2, R2 = @constinferred qr_compact!(copy!(Ac, A), (Q, R))
@@ -37,13 +37,13 @@ eltypes = (Float32, Float64, ComplexF32, ComplexF64)
         # unblocked algorithm
         qr_compact!(copy!(Ac, A), (Q, R); blocksize = 1)
         @test Q * R ≈ A
-        @test isisometry(Q)
+        @test isisometric(Q)
         qr_compact!(copy!(Ac, A), (Q2, noR); blocksize = 1)
         @test Q == Q2
         qr_compact!(copy!(Ac, A), (Q2, noR); blocksize = 1)
         qr_null!(copy!(Ac, A), N; blocksize = 1)
         @test maximum(abs, A' * N) < eps(real(T))^(2 / 3)
-        @test isisometry(N)
+        @test isisometric(N)
         if n <= m
             qr_compact!(copy!(Q2, A), (Q2, noR); blocksize = 1) # in-place Q
             @test Q ≈ Q2
@@ -54,12 +54,12 @@ eltypes = (Float32, Float64, ComplexF32, ComplexF64)
         # other blocking
         qr_compact!(copy!(Ac, A), (Q, R); blocksize = 8)
         @test Q * R ≈ A
-        @test isisometry(Q)
+        @test isisometric(Q)
         qr_compact!(copy!(Ac, A), (Q2, noR); blocksize = 8)
         @test Q == Q2
         qr_null!(copy!(Ac, A), N; blocksize = 8)
         @test maximum(abs, A' * N) < eps(real(T))^(2 / 3)
-        @test isisometry(N)
+        @test isisometric(N)
 
         # pivoted
         qr_compact!(copy!(Ac, A), (Q, R); pivoted = true)
@@ -70,21 +70,21 @@ eltypes = (Float32, Float64, ComplexF32, ComplexF64)
         # positive
         qr_compact!(copy!(Ac, A), (Q, R); positive = true)
         @test Q * R ≈ A
-        @test isisometry(Q)
+        @test isisometric(Q)
         @test all(>=(zero(real(T))), real(diag(R)))
         qr_compact!(copy!(Ac, A), (Q2, noR); positive = true)
         @test Q == Q2
         # positive and blocksize 1
         qr_compact!(copy!(Ac, A), (Q, R); positive = true, blocksize = 1)
         @test Q * R ≈ A
-        @test isisometry(Q)
+        @test isisometric(Q)
         @test all(>=(zero(real(T))), real(diag(R)))
         qr_compact!(copy!(Ac, A), (Q2, noR); positive = true, blocksize = 1)
         @test Q == Q2
         # positive and pivoted
         qr_compact!(copy!(Ac, A), (Q, R); positive = true, pivoted = true)
         @test Q * R ≈ A
-        @test isisometry(Q)
+        @test isisometric(Q)
         if n <= m
             # the following test tries to find the diagonal element (in order to test positivity)
             # before the column permutation. This only works if all columns have a diagonal
diff --git a/test/schur.jl b/test/schur.jl
index 897cf155..e24de579 100644
--- a/test/schur.jl
+++ b/test/schur.jl
@@ -14,7 +14,7 @@ using LinearAlgebra: I
         TA, Z, vals = @constinferred schur_full(A; alg)
         @test eltype(TA) == eltype(Z) == T
         @test eltype(vals) == Tc
-        @test isisometry(Z)
+        @test isisometric(Z)
         @test A * Z ≈ Z * TA
 
         Ac = similar(A)
diff --git a/test/svd.jl b/test/svd.jl
index 21d34337..a5daad96 100644
--- a/test/svd.jl
+++ b/test/svd.jl
@@ -3,7 +3,7 @@ using Test
 using TestExtras
 using StableRNGs
 using LinearAlgebra: LinearAlgebra, Diagonal, I, isposdef, norm
-using MatrixAlgebraKit: TruncatedAlgorithm, diagview, isisometry
+using MatrixAlgebraKit: TruncatedAlgorithm, diagview, isisometric
 
 const BLASFloats = (Float32, Float64, ComplexF32, ComplexF64)
 
@@ -38,8 +38,8 @@ const BLASFloats = (Float32, Float64, ComplexF32, ComplexF64)
             @test S isa Diagonal{real(T)} && size(S) == (minmn, minmn)
             @test Vᴴ isa Matrix{T} && size(Vᴴ) == (minmn, n)
             @test U * S * Vᴴ ≈ A
-            @test isisometry(U)
-            @test isisometry(Vᴴ; side = :right)
+            @test isisometric(U)
+            @test isisometric(Vᴴ; side = :right)
             @test isposdef(S)
 
             Ac = similar(A)
@@ -50,8 +50,8 @@ const BLASFloats = (Float32, Float64, ComplexF32, ComplexF64)
             @test S2 === S
             @test V2ᴴ === Vᴴ
             @test U * S * Vᴴ ≈ A
-            @test isisometry(U)
-            @test isisometry(Vᴴ; side = :right)
+            @test isisometric(U)
+            @test isisometric(Vᴴ; side = :right)
             @test isposdef(S)
 
             Sd = @constinferred svd_vals(A, alg′)

From c079d0cf8f2a57f14c2c237b660062fa446a3beb Mon Sep 17 00:00:00 2001
From: "copilot-swe-agent[bot]" <198982749+Copilot@users.noreply.github.com>
Date: Thu, 9 Oct 2025 14:21:57 +0000
Subject: [PATCH 3/6] Rename is_left_isometry and is_right_isometry to
 isleftisometric and isrightisometric

Co-authored-by: lkdvos <37111893+lkdvos@users.noreply.github.com>
---
 src/common/matrixproperties.jl | 28 ++++++++++++++--------------
 1 file changed, 14 insertions(+), 14 deletions(-)

diff --git a/src/common/matrixproperties.jl b/src/common/matrixproperties.jl
index 6f3b6e7b..feb0fb63 100644
--- a/src/common/matrixproperties.jl
+++ b/src/common/matrixproperties.jl
@@ -8,13 +8,13 @@ Test whether a linear map is an isometry, where the type of isometry is controll
 
 The `isapprox_kwargs` are passed on to `isapprox` to control the tolerances.
 
-New specializations should overload [`is_left_isometry`](@ref) and [`is_right_isometry`](@ref).
+New specializations should overload [`isleftisometric`](@ref) and [`isrightisometric`](@ref).
 
 See also [`isunitary`](@ref).
 """
 function isisometric(A; side::Symbol = :left, isapprox_kwargs...)
-    side === :left && return is_left_isometry(A; isapprox_kwargs...)
-    side === :right && return is_right_isometry(A; isapprox_kwargs...)
+    side === :left && return isleftisometric(A; isapprox_kwargs...)
+    side === :right && return isrightisometric(A; isapprox_kwargs...)
 
     throw(ArgumentError(lazy"Invalid isometry side: $side"))
 end
@@ -28,24 +28,24 @@ The `isapprox_kwargs` are passed on to `isapprox` to control the tolerances.
 See also [`isisometric`](@ref).
 """
 function isunitary(A; isapprox_kwargs...)
-    return is_left_isometry(A; isapprox_kwargs...) &&
-        is_right_isometry(A; isapprox_kwargs...)
+    return isleftisometric(A; isapprox_kwargs...) &&
+        isrightisometric(A; isapprox_kwargs...)
 end
 function isunitary(A::AbstractMatrix; isapprox_kwargs...)
     size(A, 1) == size(A, 2) || return false
-    return is_left_isometry(A; isapprox_kwargs...)
+    return isleftisometric(A; isapprox_kwargs...)
 end
 
 @doc """
-    is_left_isometry(A; isapprox_kwargs...) -> Bool
+    isleftisometric(A; isapprox_kwargs...) -> Bool
 
 Test whether a linear map is a left isometry, i.e. `A' * A ≈ I`.
 The `isapprox_kwargs` can be used to control the tolerances of the equality.
 
-See also [`isisometric`](@ref) and [`is_right_isometry`](@ref).
-""" is_left_isometry
+See also [`isisometric`](@ref) and [`isrightisometric`](@ref).
+""" isleftisometric
 
-function is_left_isometry(A::AbstractMatrix; atol::Real = 0, rtol::Real = defaulttol(A), norm = LinearAlgebra.norm)
+function isleftisometric(A::AbstractMatrix; atol::Real = 0, rtol::Real = defaulttol(A), norm = LinearAlgebra.norm)
     P = A' * A
     nP = norm(P) # isapprox would use `rtol * max(norm(P), norm(I))`
     diagview(P) .-= 1
@@ -53,15 +53,15 @@ function is_left_isometry(A::AbstractMatrix; atol::Real = 0, rtol::Real = defaul
 end
 
 @doc """
-    is_right_isometry(A; isapprox_kwargs...) -> Bool
+    isrightisometric(A; isapprox_kwargs...) -> Bool
 
 Test whether a linear map is a right isometry, i.e. `A * A' ≈ I`.
 The `isapprox_kwargs` can be used to control the tolerances of the equality.
 
-See also [`isisometric`](@ref) and [`is_left_isometry`](@ref).
-""" is_right_isometry
+See also [`isisometric`](@ref) and [`isleftisometric`](@ref).
+""" isrightisometric
 
-function is_right_isometry(A::AbstractMatrix; atol::Real = 0, rtol::Real = defaulttol(A), norm = LinearAlgebra.norm)
+function isrightisometric(A::AbstractMatrix; atol::Real = 0, rtol::Real = defaulttol(A), norm = LinearAlgebra.norm)
     P = A * A'
     nP = norm(P) # isapprox would use `rtol * max(norm(P), norm(I))`
     diagview(P) .-= 1

From 2176f58d76cd9e6063c0f543277c87f7040bd9a8 Mon Sep 17 00:00:00 2001
From: Lukas Devos <ldevos98@gmail.com>
Date: Thu, 9 Oct 2025 17:38:43 -0400
Subject: [PATCH 4/6] rename `is_left_isometric` and `is_right_isometric`

---
 src/common/matrixproperties.jl | 33 +++++++++++++++++----------------
 1 file changed, 17 insertions(+), 16 deletions(-)

diff --git a/src/common/matrixproperties.jl b/src/common/matrixproperties.jl
index feb0fb63..50c4f348 100644
--- a/src/common/matrixproperties.jl
+++ b/src/common/matrixproperties.jl
@@ -8,13 +8,14 @@ Test whether a linear map is an isometry, where the type of isometry is controll
 
 The `isapprox_kwargs` are passed on to `isapprox` to control the tolerances.
 
-New specializations should overload [`isleftisometric`](@ref) and [`isrightisometric`](@ref).
+New specializations should overload [`MatrixAlgebraKit.is_left_isometric`](@ref) and
+[`MatrixAlgebraKit.is_right_isometric`](@ref).
 
 See also [`isunitary`](@ref).
 """
 function isisometric(A; side::Symbol = :left, isapprox_kwargs...)
-    side === :left && return isleftisometric(A; isapprox_kwargs...)
-    side === :right && return isrightisometric(A; isapprox_kwargs...)
+    side === :left && return is_left_isometric(A; isapprox_kwargs...)
+    side === :right && return is_right_isometric(A; isapprox_kwargs...)
 
     throw(ArgumentError(lazy"Invalid isometry side: $side"))
 end
@@ -28,24 +29,24 @@ The `isapprox_kwargs` are passed on to `isapprox` to control the tolerances.
 See also [`isisometric`](@ref).
 """
 function isunitary(A; isapprox_kwargs...)
-    return isleftisometric(A; isapprox_kwargs...) &&
-        isrightisometric(A; isapprox_kwargs...)
+    return is_left_isometric(A; isapprox_kwargs...) &&
+        is_right_isometric(A; isapprox_kwargs...)
 end
 function isunitary(A::AbstractMatrix; isapprox_kwargs...)
     size(A, 1) == size(A, 2) || return false
-    return isleftisometric(A; isapprox_kwargs...)
+    return is_left_isometric(A; isapprox_kwargs...)
 end
 
 @doc """
-    isleftisometric(A; isapprox_kwargs...) -> Bool
+    is_left_isometric(A; isapprox_kwargs...) -> Bool
 
-Test whether a linear map is a left isometry, i.e. `A' * A ≈ I`.
+Test whether a linear map is a (left) isometry, i.e. `A' * A ≈ I`.
 The `isapprox_kwargs` can be used to control the tolerances of the equality.
 
-See also [`isisometric`](@ref) and [`isrightisometric`](@ref).
-""" isleftisometric
+See also [`isisometric`](@ref) and [`MatrixAlgebraKit.is_right_isometric`](@ref).
+""" is_left_isometric
 
-function isleftisometric(A::AbstractMatrix; atol::Real = 0, rtol::Real = defaulttol(A), norm = LinearAlgebra.norm)
+function is_left_isometric(A::AbstractMatrix; atol::Real = 0, rtol::Real = defaulttol(A), norm = LinearAlgebra.norm)
     P = A' * A
     nP = norm(P) # isapprox would use `rtol * max(norm(P), norm(I))`
     diagview(P) .-= 1
@@ -53,15 +54,15 @@ function isleftisometric(A::AbstractMatrix; atol::Real = 0, rtol::Real = default
 end
 
 @doc """
-    isrightisometric(A; isapprox_kwargs...) -> Bool
+    is_right_isometric(A; isapprox_kwargs...) -> Bool
 
-Test whether a linear map is a right isometry, i.e. `A * A' ≈ I`.
+Test whether a linear map is a (right) isometry, i.e. `A * A' ≈ I`.
 The `isapprox_kwargs` can be used to control the tolerances of the equality.
 
-See also [`isisometric`](@ref) and [`isleftisometric`](@ref).
-""" isrightisometric
+See also [`isisometric`](@ref) and [`MatrixAlgebraKit.is_left_isometric`](@ref).
+""" is_right_isometric
 
-function isrightisometric(A::AbstractMatrix; atol::Real = 0, rtol::Real = defaulttol(A), norm = LinearAlgebra.norm)
+function is_right_isometric(A::AbstractMatrix; atol::Real = 0, rtol::Real = defaulttol(A), norm = LinearAlgebra.norm)
     P = A * A'
     nP = norm(P) # isapprox would use `rtol * max(norm(P), norm(I))`
     diagview(P) .-= 1

From 1e36ef1c52d8eefd3110c88ef2c7ba76bb6dc54b Mon Sep 17 00:00:00 2001
From: Lukas Devos <ldevos98@gmail.com>
Date: Thu, 9 Oct 2025 17:38:47 -0400
Subject: [PATCH 5/6] mark as public

---
 src/MatrixAlgebraKit.jl | 1 +
 1 file changed, 1 insertion(+)

diff --git a/src/MatrixAlgebraKit.jl b/src/MatrixAlgebraKit.jl
index 86c5ef9f..5211476f 100644
--- a/src/MatrixAlgebraKit.jl
+++ b/src/MatrixAlgebraKit.jl
@@ -65,6 +65,7 @@ export notrunc, truncrank, trunctol, truncerror, truncfilter
             :svd_pullback!, :svd_trunc_pullback!
         )
     )
+    eval(Expr(:public, :is_left_isometric, :is_right_isometric))
 end
 
 include("common/defaults.jl")

From dc17ad718280cf40d20963cb6cc82a13710bd497 Mon Sep 17 00:00:00 2001
From: Lukas Devos <ldevos98@gmail.com>
Date: Thu, 9 Oct 2025 19:29:55 -0400
Subject: [PATCH 6/6] refactor is_right_isometric

---
 src/common/matrixproperties.jl | 8 +-------
 1 file changed, 1 insertion(+), 7 deletions(-)

diff --git a/src/common/matrixproperties.jl b/src/common/matrixproperties.jl
index 50c4f348..224a16cd 100644
--- a/src/common/matrixproperties.jl
+++ b/src/common/matrixproperties.jl
@@ -61,13 +61,7 @@ The `isapprox_kwargs` can be used to control the tolerances of the equality.
 
 See also [`isisometric`](@ref) and [`MatrixAlgebraKit.is_left_isometric`](@ref).
 """ is_right_isometric
-
-function is_right_isometric(A::AbstractMatrix; atol::Real = 0, rtol::Real = defaulttol(A), norm = LinearAlgebra.norm)
-    P = A * A'
-    nP = norm(P) # isapprox would use `rtol * max(norm(P), norm(I))`
-    diagview(P) .-= 1
-    return norm(P) <= max(atol, rtol * nP) # assume that the norm of I is `sqrt(n)`
-end
+is_right_isometric(A; kwargs...) = is_left_isometric(A'; kwargs...)
 
 """
     ishermitian(A; isapprox_kwargs...)