diff --git a/src/common/matrixproperties.jl b/src/common/matrixproperties.jl
index 25876b90..9180ae15 100644
--- a/src/common/matrixproperties.jl
+++ b/src/common/matrixproperties.jl
@@ -50,7 +50,7 @@ function is_left_isometric(A::AbstractMatrix; atol::Real = 0, rtol::Real = defau
     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)`
+    return norm(P) ≤ max(atol, rtol * nP) # assume that the norm of I is `sqrt(n)`
 end
 
 @doc """
diff --git a/test/ad_utils.jl b/test/ad_utils.jl
index 4c03e50c..7a7cf39a 100644
--- a/test/ad_utils.jl
+++ b/test/ad_utils.jl
@@ -27,5 +27,36 @@ function remove_eighgauge_dependence!(
     mul!(ΔV, V, gaugepart, -1, 1)
     return ΔV
 end
+function stabilize_eigvals!(D::AbstractVector)
+    absD = abs.(D)
+    p = invperm(sortperm(absD)) # rank of abs(D)
+    # account for exact degeneracies in absolute value when having complex conjugate pairs
+    for i in 1:(length(D) - 1)
+        if absD[i] == absD[i + 1] # conjugate pairs will appear sequentially
+            p[p .>= p[i + 1]] .-= 1 # lower the rank of all higher ones
+        end
+    end
+    n = maximum(p)
+    # rescale eigenvalues so that they lie on distinct radii in the complex plane
+    # that are chosen randomly in non-overlapping intervals [k/n, (k+0.5)/n)] for k=1,...,n
+    radii = ((1:n) .+ rand(real(eltype(D)), n) ./ 2) ./ n
+    for i in 1:length(D)
+        D[i] = sign(D[i]) * radii[p[i]]
+    end
+    return D
+end
+function make_eig_matrix(rng, T, n)
+    A = randn(rng, T, n, n)
+    D, V = eig_full(A)
+    stabilize_eigvals!(diagview(D))
+    Ac = V * D * inv(V)
+    return (T <: Real) ? real(Ac) : Ac
+end
+function make_eigh_matrix(rng, T, n)
+    A = project_hermitian!(randn(rng, T, n, n))
+    D, V = eigh_full(A)
+    stabilize_eigvals!(diagview(D))
+    return project_hermitian!(V * D * V')
+end
 
 precision(::Type{T}) where {T <: Number} = sqrt(eps(real(T)))
diff --git a/test/chainrules.jl b/test/chainrules.jl
index 4be77380..a8b2fd3b 100644
--- a/test/chainrules.jl
+++ b/test/chainrules.jl
@@ -221,7 +221,7 @@ end
     rng = StableRNG(12345)
     m = 19
     atol = rtol = m * m * precision(T)
-    A = randn(rng, T, m, m)
+    A = make_eig_matrix(rng, T, m)
     D, V = eig_full(A)
     Ddiag = diagview(D)
     ΔV = randn(rng, complex(T), m, m)
@@ -297,8 +297,7 @@ end
     rng = StableRNG(12345)
     m = 19
     atol = rtol = m * m * precision(T)
-    A = randn(rng, T, m, m)
-    A = A + A'
+    A = make_eigh_matrix(rng, T, m)
     D, V = eigh_full(A)
     Ddiag = diagview(D)
     ΔV = randn(rng, T, m, m)
diff --git a/test/mooncake.jl b/test/mooncake.jl
index 0c1b5d56..760102b1 100644
--- a/test/mooncake.jl
+++ b/test/mooncake.jl
@@ -238,7 +238,7 @@ end
     rng = StableRNG(12345)
     m = 19
     atol = rtol = m * m * precision(T)
-    A = randn(rng, T, m, m)
+    A = make_eig_matrix(rng, T, m)
     DV = eig_full(A)
     D, V = DV
     Ddiag = diagview(D)
@@ -347,9 +347,9 @@ MatrixAlgebraKit.copy_input(::typeof(copy_eigh_trunc_no_error), A) = MatrixAlgeb
     rng = StableRNG(12345)
     m = 19
     atol = rtol = m * m * precision(T)
-    A = randn(rng, T, m, m)
-    A = A + A'
+    A = make_eigh_matrix(rng, T, m)
     D, V = eigh_full(A)
+    Ddiag = diagview(D)
     ΔV = randn(rng, T, m, m)
     ΔV = remove_eighgauge_dependence!(ΔV, D, V; degeneracy_atol = atol)
     ΔD = randn(rng, real(T), m, m)
@@ -357,7 +357,6 @@ MatrixAlgebraKit.copy_input(::typeof(copy_eigh_trunc_no_error), A) = MatrixAlgeb
     dD = make_mooncake_tangent(ΔD2)
     dV = make_mooncake_tangent(ΔV)
     dDV = Mooncake.build_tangent(typeof((ΔD2, ΔV)), dD, dV)
-    Ddiag = diagview(D)
     @testset for alg in (
             LAPACK_QRIteration(),
             #LAPACK_DivideAndConquer(),