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A Sharp Unitarily Invariant Norm Bound for the Off-Diagonal Block Perturbation of a Hermitian Matrix
Lei-Hong Zhang, Ren-Cang Li
TL;DR
The paper asks whether Li and Li’s sharp spectral-norm eigenvalue perturbation bound extends to total eigenvalue changes measured by unitarily invariant norms. It develops a norm- and eigenvalue-path framework, proving the extension for Q-norms and for rank(E)≤1. The unrestricted unitarily invariant-norm case remains open.
Problem
The paper investigates whether the spectral-norm bound for eigenvalue changes extends to some or all unitarily invariant norms.
Method
The paper uses symmetric-gauge and eigenvalue-path arguments to analyze the diagonal matrix of ordered eigenvalue differences under the block perturbation.
Results
The extension holds for Q-norms, including Schatten p-norms for 2≤p≤∞, and for any unitarily invariant norm when rank(E)≤1.
Takeaways & Limitations
The result covers the Frobenius and spectral norms and provides a broader perturbation bound while leaving the fully unrestricted norm case unresolved.
Takeaways & Limitations
Whether the extension holds for every unitarily invariant norm without restricting rank(E) remains open.
Abstract
from arXiv · showhide
Let $$ A=\begin{bmatrix} H_1 & E^* \\ E & H_2 \end{bmatrix} \quad\text{and}\quad \widetilde A=\begin{bmatrix} H_1 & 0 \\ 0 & H_2 \end{bmatrix} $$ be two partitioned Hermitian matrices, where $\widetilde A$ is obtained from $A$ by simply dropping the off-diagonal blocks, and let $η$ be the gap between the spectra ${\rm eig}(H_1)$ of $H_1$ and ${\rm eig}(H_2)$ of $H_2$. Define, for $δ\ge 0$ and $ε\ge 0$, $$ φ(δ,ε)= \begin{cases} 2ε/(δ+\sqrt{δ^2+4ε^2}), &\quad\mbox{if $(δ,ε)\ne (0,0)$}, 1, &\quad\mbox{if $(δ,ε) = (0,0)$}, \end{cases} $$ and let $V=A-\widetilde A$ and $ε_2=\|E\|_2=\|V\|_2$, the matrix spectral norm. Li and Li [{\em Linear Algebra Appl.}, 395:183--190, 2005] established a sharp spectral-norm bound on the changes in the eigenvalues of $A$: $$ \big\|{\rm diag}\big(\pmbλ(A)-\pmbλ(\widetilde A)\big)\big\|_2 \le φ(η,ε_2)\,\|E\|_2, $$ where $\pmbλ(A)$ is the vector whose components are the eigenvalues of $A$ in descending order and similarly for $\pmbλ(\widetilde A)$. The goal of this paper is to resolve the question: how far an extension of this result in the form $$ \big\|{\rm diag}\big(\pmbλ(A)-\pmbλ(\widetilde A)\big)\big\|_{\rm UI} \le φ(η,ε_2)\,\|A-\widetilde A\|_{\rm UI} $$ remains valid for some or all unitarily invariant norms $\|\cdot\|_{\rm UI}$? Two results are obtained: (a) the extension holds for any $Q$-norm, a subclass of unitarily invariant norms that encompasses the Schatten $p$-norm for $2\le p\le\infty$ (particularly, the Frobenius norm and the spectral norm included), and (b) the extension holds for any unitarily invariant norm if ${\rm rank}(E)\le 1$. It is demonstrated that the equality is attained on the $2\times 2$ matrix $A$.
1 Introduction
The paper asks whether Li and Li’s sharp spectral-norm eigenvalue perturbation bound extends to unitarily invariant norms. It proves the extension for Q-norms and, without rank restrictions on the norm, for rank(E)≤1, while leaving the unrestricted case open.
- Motivation: The problem studies eigenvalue changes when the off-diagonal blocks E and E* are dropped, reducing A to independent problems for H1 and H2.The goal is to bound the resulting eigenvalue changes using unitarily invariant norms.
- Existing bound: Li and Li’s bound controls each difference |λi(A)−λi(Ã)| by ϕ(ηi, ε2)ε2, with ηi the individual spectral gap.The related bound using the common gap η follows because ηi≥η.
- Existing bound: The bound combines linear ε2 behavior with quadratic ε2^2/η behavior, avoiding the latter’s blow-up when η is small.The quadratic bound can be smaller when η is sufficiently large, but becomes problematic as η approaches zero.
- Research question: The paper asks whether analogous total eigenvalue-change bounds hold for some or all unitarily invariant norms.The comparison is framed through the diagonal matrix of ordered eigenvalue differences.
- Contributions: The extension holds for any Q-norm, including Schatten p-norms for 2≤p≤∞, especially the Frobenius and spectral norms.It also holds for every unitarily invariant norm when rank(E)≤1.
- Limitations: Whether the extension holds for every unitarily invariant norm without restricting rank(E) remains open.The paper also notes that its sections address preliminary results, Q-norms, rank-one coupling, singular values, and conclusions.
2 Preliminaries
The preliminaries connect the block perturbation to singular values of E and develop a differentiable eigenvalue-path framework. They also establish the regularity and integration tools used to control ordered eigenvalue changes.
- Integration framework: The path satisfies λ(0)=λ(Ã) and λ(1)=λ(A), so integrating the derivative representation connects the two ordered spectra.The integration lemma applies to Lipschitz, hence absolutely continuous, vector-valued paths.
- Block perturbation: The nonzero singular values of V are those of E, each repeated twice.When rank(E)≤1, V has two singular values equal to ε2 and N−2 zero singular values.
- General perturbation tool: The key lemma applies to arbitrary Hermitian à and V, not only to the paper’s block-structured matrices.This broadens the perturbation-path tool beyond the immediate block setting.
- Eigenvalue paths: For H(t)=Ã+tV, the ordered eigenvalue functions λi(t) are Lipschitz continuous and differentiable almost everywhere.At differentiability points, suitable eigenvectors can be selected even when eigenvalues have multiplicities.
- Eigenvalue paths: At almost every t, eigenvalue derivatives coincide as a multiset with Rayleigh quotients xj(t)*Vxj(t) from a suitably chosen eigenbasis.Within repeated eigenspaces, the basis is chosen by diagonalizing the compression of V.
3 The Q-norm
The Q-norm section establishes the extension of the eigenvalue perturbation bound from the spectral norm to all Q-norms, using symmetric gauge functions and majorization. This includes every Schatten p-norm for 2 ≤ p ≤ ∞, with equality attained by a 2 × 2 example.
- Q-norm framework: Q-norms are unitarily invariant norms representable through another unitarily invariant norm applied to Y*Y and a square root.The section also uses the one-to-one correspondence between unitarily invariant norms and symmetric gauge functions.
- Consequences: The Q-norm class contains all Schatten p-norms for 2 ≤ p ≤ ∞, including the Frobenius and spectral norms.Thus the theorem yields the Frobenius-norm specialization as a new bound while recovering the spectral-norm result.
- Proof strategy: The proof controls eigenvalue derivatives through the spectral gap η and the factor c(t), then transfers componentwise bounds to unitarily invariant norms via majorization.The block structure of V links the derivative term to E and the singular values of V.
- Sharpness: Equality in (1.10) is attained by the 2 × 2 example with m = n = 1, scalar diagonal blocks, and scalar E.Because this example has rank(E) = 1, the equality also extends to every unitarily invariant norm in the later result.
4 Case rank(E) ≤1
For rank(E) ≤ 1, the paper extends the perturbation inequality to every unitarily invariant norm. The proof converts projector-based estimates into weak majorization, which is equivalent to domination under all symmetric gauge functions.
- Main result: Theorem 4.1 proves inequality (1.10) for every unitarily invariant norm when rank(E) ≤ 1.The rank-zero case is trivial, so the substantive argument treats rank(E) = 1.
- Rank-one structure: Rank-one structure gives V two singular values equal to ε2 and N − 2 zero singular values.This low-rank spectrum is used to control cumulative eigenvalue differences.
- Proof strategy: The proof uses orthogonal projectors commuting with H(t) to bound grouped eigenvalue-derivative contributions.Projector compressions and trace identities exploit the rank-one block structure and the spectral gap.
- Majorization step: Weak majorization of the sorted absolute eigenvalue changes by the corresponding bound implies the result for every unitarily invariant norm.Fan’s dominance theorem supplies the equivalence between weak majorization and domination by all symmetric gauge functions.
- Sharpness: The 2 × 2 example attains equality in (1.10) for every unitarily invariant norm.This example simultaneously has rank(E) = 1 and demonstrates sharpness of the all-norm extension.
5 Application to Singular Value Problem
The paper applies the Q-norm theorem to singular-value perturbation bounds. The resulting estimate improves the classical Mirsky bound for Q-norms when the relevant spectral gap is positive and the off-diagonal blocks are small.
- Application: The Q-norm eigenvalue result applies directly to the singular-value problem.The construction embeds the singular values into eigenvalues of related Hermitian block matrices.
- Specialization: The Frobenius norm gives a direct specialization of the singular-value bound.The paper identifies this specialization as a consequence of the general Q-norm result.
- Bound: For any Q-norm, the squared singular-value perturbation inequality follows from the corresponding Hermitian eigenvalue bound.The derivation yields ∥X*X∥ui ≤ [φ(η, ε2)]^2∥Y*Y∥ui.
- Comparison: For Q-norms, the new inequality improves Mirsky’s bound because φ(η, ε2) ∈ [0, 1] and may be much smaller than 1 for positive η and small coupling blocks.The improvement is stated relative to the classical unitarily invariant-norm SVD perturbation theorem.
6 Conclusion
The paper extends the Li–Li perturbation bound to selected unitarily invariant norms, proving cases for Q-norms and rank-one off-diagonal blocks while leaving the unrestricted case open.
- 6 Conclusion: The Li–Li bound combines the Weyl–Lidskii theorem with a quadratic residual bound into a sharper perturbation inequality for block Hermitian matrices.The bound concerns eigenvalue changes after dropping the off-diagonal blocks.
- 6 Conclusion: The extension holds for Q-norms, including Schatten p-norms for 2 ≤ p ≤ ∞, the Frobenius norm, and the spectral norm.Q-norms form a subclass of unitarily invariant norms.
- 6 Conclusion: The extension holds for every unitarily invariant norm when rank(E) ≤ 1, including cases where one diagonal block is scalar.The scalar-block case occurs when one of H1 and H2 is 1-by-1.
- 6 Conclusion: An application to the singular value problem is also made.
- 6 Conclusion: Whether the extension holds for every unitarily invariant norm without a rank restriction remains open.
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The authors disclose using AI tools to develop ideas presented in the paper and retain responsibility for all content.
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