The Davis–Kahan sin Θ theorems in unitarily invariant seminorms

Let A be a symmetric operator on a finite-dimensional inner product space E and U an A-invariant subspace. This file bounds, in every unitarily invariant seminorm N, how far a second subspace sticks out of U, under an interval/exterior separation of spectra. The bounds have constant 1.

The residual form

Let M be a symmetric operator on a second finite-dimensional space F and X : F → E any linear map, with residual R = A X - X M. If the eigenvalues of M lie in [a, b] and those of A on Uᗮ lie outside (a - δ, b + δ), then

δ N(P_{Uᗮ} X) ≤ N(R).

The projection P_{Uᗮ} commutes with A, so Y = P_{Uᗮ} X solves the Sylvester equation A Y - Y M = P_{Uᗮ} R. Since Y takes values in Uᗮ, the interval/exterior Sylvester bound on invariant subspaces gives δ N(Y) ≤ N(P_{Uᗮ} R) ≤ N(R). When X is an isometry with range Z, P_{Uᗮ} X is the directed sine map from Z towards U; when moreover M = X† A X is the compression of A to Z, R is the Ritz residual.

The perturbation form

Let B be a second symmetric operator on E and V a B-invariant subspace. If the eigenvalues of A on U lie in [a, b] and those of B on Vᗮ lie outside (a - δ, b + δ), then the directed sine map P_{Vᗮ} P_U from U towards V satisfies

δ N(P_{Vᗮ} P_U) ≤ N(B - A),

and the same holds with the roles of the interval and its exterior exchanged. Here Y = P_{Vᗮ} P_U takes values in Vᗮ, vanishes on Uᗮ, and solves B Y - Y A = P_{Vᗮ} (B - A) P_U, so the interval/exterior Sylvester bound on invariant subspaces gives δ N(Y) ≤ N(P_{Vᗮ} (B - A) P_U) ≤ N(B - A). The dimension-free operator-norm form under quadratic-form separation is Submodule.mul_norm_starProjection_comp_starProjection_le_norm_sub.

Main results

• TauCeti.UnitarilyInvariantSeminorm.mul_map_starProjection_comp_le_map_sylvester_apply: the residual sin Θ theorem δ N(P_{Uᗮ} X) ≤ N(A X - X M).
• TauCeti.UnitarilyInvariantSeminorm.mul_map_starProjection_comp_starProjection_le_map_sub, TauCeti.UnitarilyInvariantSeminorm.mul_map_starProjection_comp_starProjection_le_map_sub_swap: the perturbation sin Θ theorem δ N(P_{Vᗮ} P_U) ≤ N(B - A), in both orientations of the interval/exterior separation.
• TauCeti.UnitarilyInvariantSeminorm.map_sylvester_starProjection_comp_starProjection_le: on the directed sine map, N(B Y - Y A) ≤ N(B - A), which turns every Sylvester bound into a perturbation sin Θ bound.

References

• C. Davis, W. M. Kahan, The rotation of eigenvectors by a perturbation. III, SIAM J. Numer. Anal. 7 (1970), 1–46.
• R. Bhatia, Matrix Analysis, Graduate Texts in Mathematics 169, Springer (1997), §VII.3.

theorem TauCeti.UnitarilyInvariantSeminorm.mul_map_starProjection_comp_le_map_sylvester_apply {𝕜 : Type u_1} {E : Type u_2} {F : Type u_3} [RCLike 𝕜] [NormedAddCommGroup E] [InnerProductSpace 𝕜 E] [FiniteDimensional 𝕜 E] [NormedAddCommGroup F] [InnerProductSpace 𝕜 F] [FiniteDimensional 𝕜 F] {a b δ : ℝ} (N : UnitarilyInvariantSeminorm 𝕜 F E) {A : E →L[𝕜] E} {M : F →L[𝕜] F} (hA : (↑A).IsSymmetric) (hM : (↑M).IsSymmetric) {U : Submodule 𝕜 E} (hU : ∀ x ∈ U, A x ∈ U) (h : Module.End.PointIntervalExteriorGap ↑M ⊤ (↑A) Uᗮ a b δ) (X : F →L[𝕜] E) : δ * N ↑(Uᗮ.starProjection ∘SL X) ≤ N ↑((A.sylvester M) X)

The residual Davis–Kahan sin Θ theorem. Let A and M be symmetric operators on finite-dimensional inner product spaces E and F, let U be invariant under A, and let X : F → E be linear. If the eigenvalues of M lie in [a, b] and those of A on Uᗮ lie outside (a - δ, b + δ), then δ N(P_{Uᗮ} X) ≤ N(A X - X M) for every unitarily invariant seminorm N.

theorem TauCeti.UnitarilyInvariantSeminorm.map_sylvester_starProjection_comp_starProjection_le {𝕜 : Type u_1} {E : Type u_2} [RCLike 𝕜] [NormedAddCommGroup E] [InnerProductSpace 𝕜 E] [FiniteDimensional 𝕜 E] (N : UnitarilyInvariantSeminorm 𝕜 E E) {A B : E →L[𝕜] E} (hA : (↑A).IsSymmetric) (hB : (↑B).IsSymmetric) {U V : Submodule 𝕜 E} (hU : ∀ x ∈ U, A x ∈ U) (hV : ∀ x ∈ V, B x ∈ V) : N ↑((B.sylvester A) (Vᗮ.starProjection ∘SL U.starProjection)) ≤ N (↑B - ↑A)

On the directed sine map P_{Vᗮ} P_U from an A-invariant subspace U towards a B-invariant subspace V, the Sylvester operator Y ↦ B Y - Y A is P_{Vᗮ} (B - A) P_U, so its unitarily invariant seminorm is at most N(B - A).

theorem TauCeti.UnitarilyInvariantSeminorm.mul_map_starProjection_comp_starProjection_le_map_sub {𝕜 : Type u_1} {E : Type u_2} [RCLike 𝕜] [NormedAddCommGroup E] [InnerProductSpace 𝕜 E] [FiniteDimensional 𝕜 E] {a b δ : ℝ} (N : UnitarilyInvariantSeminorm 𝕜 E E) {A B : E →L[𝕜] E} (hA : (↑A).IsSymmetric) (hB : (↑B).IsSymmetric) {U V : Submodule 𝕜 E} (hU : ∀ x ∈ U, A x ∈ U) (hV : ∀ x ∈ V, B x ∈ V) (h : Module.End.PointIntervalExteriorGap (↑A) U (↑B) Vᗮ a b δ) : δ * N ↑(Vᗮ.starProjection ∘SL U.starProjection) ≤ N (↑B - ↑A)

The perturbation Davis–Kahan sin Θ theorem. Let A and B be symmetric operators on a finite-dimensional inner product space, let U be invariant under A and V invariant under B. If the eigenvalues of A on U lie in [a, b] and those of B on Vᗮ lie outside (a - δ, b + δ), then the directed sine map P_{Vᗮ} P_U from U towards V satisfies δ N(P_{Vᗮ} P_U) ≤ N(B - A) for every unitarily invariant seminorm N.

theorem TauCeti.UnitarilyInvariantSeminorm.mul_map_starProjection_comp_starProjection_le_map_sub_swap {𝕜 : Type u_1} {E : Type u_2} [RCLike 𝕜] [NormedAddCommGroup E] [InnerProductSpace 𝕜 E] [FiniteDimensional 𝕜 E] {a b δ : ℝ} (N : UnitarilyInvariantSeminorm 𝕜 E E) {A B : E →L[𝕜] E} (hA : (↑A).IsSymmetric) (hB : (↑B).IsSymmetric) {U V : Submodule 𝕜 E} (hU : ∀ x ∈ U, A x ∈ U) (hV : ∀ x ∈ V, B x ∈ V) (h : Module.End.PointIntervalExteriorGap (↑B) Vᗮ (↑A) U a b δ) : δ * N ↑(Vᗮ.starProjection ∘SL U.starProjection) ≤ N (↑B - ↑A)

The perturbation Davis–Kahan sin Θ theorem, reversed orientation. Let A and B be symmetric operators on a finite-dimensional inner product space, let U be invariant under A and V invariant under B. If the eigenvalues of B on Vᗮ lie in [a, b] and those of A on U lie outside (a - δ, b + δ), then the directed sine map P_{Vᗮ} P_U from U towards V satisfies δ N(P_{Vᗮ} P_U) ≤ N(B - A) for every unitarily invariant seminorm N.