Update and use MatrixAlgebraKit.isisometry

Correctly implement `isisometry`

Author: lkdvos

src/TensorKit.jl

@@ -73,7 +73,7 @@ export mul!, lmul!, rmul!, adjoint!, pinv, axpy!, axpby!
 export leftorth, rightorth, leftnull, rightnull, leftpolar, rightpolar,
        leftorth!, rightorth!, leftnull!, rightnull!, leftpolar!, rightpolar!,
        tsvd!, tsvd, eigen, eigen!, eig, eig!, eigh, eigh!, exp, exp!,
-       isposdef, isposdef!, ishermitian, isisometry, sylvester, rank, cond
+       isposdef, isposdef!, ishermitian, isisometry, isunitary, sylvester, rank, cond
 export braid, braid!, permute, permute!, transpose, transpose!, twist, twist!, repartition,
        repartition!
 export catdomain, catcodomain

src/auxiliary/linalg.jl

@@ -84,8 +84,6 @@ end
 safesign(s::Real) = ifelse(s < zero(s), -one(s), +one(s))
 safesign(s::Complex) = ifelse(iszero(s), one(s), s / abs(s))
 
-isisometry(A::StridedMatrix; kwargs...) = isapprox(A' * A, LinearAlgebra.I, kwargs...)
-
 function leftorth!(A::StridedMatrix{<:BlasFloat}, alg::Union{QR,QRpos}, atol::Real)
     iszero(atol) || throw(ArgumentError("nonzero atol not supported by $alg"))
     m, n = size(A)

src/tensors/factorizations/factorizations.jl

@@ -174,12 +174,15 @@ function LinearAlgebra.isposdef!(t::TensorMap)
 end
 
 # TODO: tolerances are per-block, not global or weighted - does that matter?
-function isisometry(t::AbstractTensorMap; kwargs...)
+function MatrixAlgebraKit.is_left_isometry(t::AbstractTensorMap; kwargs...)
     domain(t) ≾ codomain(t) || return false
-    for (_, b) in blocks(t)
-        MatrixAlgebra.isisometry(b; kwargs...) || return false
-    end
-    return true
+    f((c, b)) = MatrixAlgebraKit.is_left_isometry(b; kwargs...)
+    return all(f, blocks(t))
+end
+function MatrixAlgebraKit.is_right_isometry(t::AbstractTensorMap; kwargs...)
+    domain(t) ≿ codomain(t) || return false
+    f((c, b)) = MatrixAlgebraKit.is_right_isometry(b; kwargs...)
+    return all(f, blocks(t))
 end
 
 end

test/tensors.jl

@@ -365,9 +365,8 @@ for V in spacelist
             for T in (Float64, ComplexF64)
                 t1 = randisometry(T, W1, W2)
                 t2 = randisometry(T, W2 ← W2)
-                @test t1' * t1 ≈ one(t2)
-                @test t2' * t2 ≈ one(t2)
-                @test t2 * t2' ≈ one(t2)
+                @test isisometry(t1)
+                @test isunitary(t2)
                 P = t1 * t1'
                 @test P * P ≈ P
             end
@@ -447,21 +446,14 @@ for V in spacelist
                                                        TensorKit.Polar(), TensorKit.SVD(),
                                                        TensorKit.SDD())
                         Q, R = @constinferred leftorth(t, ((3, 4, 2), (1, 5)); alg=alg)
-                        QdQ = Q' * Q
-                        @test QdQ ≈ one(QdQ)
+                        @test isisometry(Q)
                         @test Q * R ≈ permute(t, ((3, 4, 2), (1, 5)))
-                        # removed since leftorth now merges legs!
-                        # if alg isa Polar
-                        #     @test isposdef(R)
-                        #     @test domain(R) == codomain(R) == space(t, 1)' ⊗ space(t, 5)'
-                        # end
                     end
                     @testset "leftnull with $alg" for alg in
                                                       (TensorKit.QR(), TensorKit.SVD(),
                                                        TensorKit.SDD())
                         N = @constinferred leftnull(t, ((3, 4, 2), (1, 5)); alg=alg)
-                        NdN = N' * N
-                        @test NdN ≈ one(NdN)
+                        @test isisometry(N)
                         @test norm(N' * permute(t, ((3, 4, 2), (1, 5)))) <
                               100 * eps(norm(t))
                     end
@@ -471,30 +463,21 @@ for V in spacelist
                                                         TensorKit.Polar(), TensorKit.SVD(),
                                                         TensorKit.SDD())
                         L, Q = @constinferred rightorth(t, ((3, 4), (2, 1, 5)); alg=alg)
-                        QQd = Q * Q'
-                        @test QQd ≈ one(QQd)
+                        @test isisometry(Q; side=:right)
                         @test L * Q ≈ permute(t, ((3, 4), (2, 1, 5)))
-                        # removed since rightorth now merges legs!
-                        # if alg isa Polar
-                        #     @test isposdef(L)
-                        #     @test domain(L) == codomain(L) == space(t, 3) ⊗ space(t, 4)
-                        # end
                     end
                     @testset "rightnull with $alg" for alg in
                                                        (TensorKit.LQ(), TensorKit.SVD(),
                                                         TensorKit.SDD())
                         M = @constinferred rightnull(t, ((3, 4), (2, 1, 5)); alg=alg)
-                        MMd = M * M'
-                        @test MMd ≈ one(MMd)
+                        @test isisometry(M; side=:right)
                         @test norm(permute(t, ((3, 4), (2, 1, 5))) * M') <
                               100 * eps(norm(t))
                     end
                     @testset "tsvd with $alg" for alg in (TensorKit.SVD(), TensorKit.SDD())
                         U, S, V = @constinferred tsvd(t, ((3, 4, 2), (1, 5)); alg=alg)
-                        UdU = U' * U
-                        @test UdU ≈ one(UdU)
-                        VVd = V * V'
-                        @test VVd ≈ one(VVd)
+                        @test isisometry(U)
+                        @test isisometry(V; side=:right)
                         t2 = permute(t, ((3, 4, 2), (1, 5)))
                         @test U * S * V ≈ t2
 
@@ -537,8 +520,7 @@ for V in spacelist
                                                       (TensorKit.QR(), TensorKit.SVD(),
                                                        TensorKit.SDD())
                         N = @constinferred leftnull(t; alg=alg)
-                        @test N' * N ≈ id(domain(N))
-                        @test N * N' ≈ id(codomain(N))
+                        @test isunitary(N)
                     end
                     @testset "rightorth with $alg" for alg in
                                                        (TensorKit.RQ(), TensorKit.RQpos(),
@@ -553,8 +535,7 @@ for V in spacelist
                                                        (TensorKit.LQ(), TensorKit.SVD(),
                                                         TensorKit.SDD())
                         M = @constinferred rightnull(copy(t'); alg=alg)
-                        @test M * M' ≈ id(codomain(M))
-                        @test M' * M ≈ id(domain(M))
+                        @test isunitary(M)
                     end
                     @testset "tsvd with $alg" for alg in (TensorKit.SVD(), TensorKit.SDD())
                         U, S, V = @constinferred tsvd(t; alg=alg)
@@ -590,8 +571,7 @@ for V in spacelist
                     @test !isposdef(t2) # unlikely for non-hermitian map
                     t2 = (t2 + t2')
                     D, V = eigen(t2)
-                    VdV = V' * V
-                    @test VdV ≈ one(VdV)
+                    @test isisometry(V)
                     D̃, Ṽ = @constinferred eigh(t2)
                     @test D ≈ D̃
                     @test V ≈ Ṽ