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Variational Analysis in Spectral Decomposition Systems

Hòa T. Bùi, Minh N. Bùi, Christian Clason

TL;DR

The paper develops a unified variational analysis for spectral functions and spectral sets that depend only on eigenvalues or singular values by formalizing spectral decomposition systems. It derives comprehensive representations of Fréchet and limiting normal cones and subdifferentials, and establishes Fréchet differentiability transfer and Clarke subdifferentials in terms of invariant functions. A key contribution is a generalization of Lidskii's theorem that describes spectrum perturbations within arbitrary spectral decomposition systems. The framework subsumes eigenvalue and singular value settings, normal decomposition systems, and Euclidean Jordan algebras, enabling streamlined, geometry-based derivations of optimality conditions for matrix problems. Overall, the results unify disparate spectral analyses and open avenues for explicit optimality conditions and Lipschitz-like properties in spectral optimization problems.

Abstract

This work is concerned with variational analysis of so-called spectral functions and spectral sets of matrices that only depend on eigenvalues of the matrix. Based on our previous work [H. T. Bùi, M. N. Bùi, and C. Clason, Convex analysis in spectral decomposition systems, arXiv 2503.14981] on convex analysis of such functions, we consider the question in the abstract framework of spectral decomposition systems, which covers a wide range of previously studied settings, including eigenvalue decomposition of Hermitian matrices and singular value decomposition of rectangular matrices, and allows deriving new results in more general settings such as normal decomposition systems and signed singular value decompositions. The main results characterize Fréchet and limiting normal cones to spectral sets as well as Fréchet, limiting, and Clarke subdifferentials of spectral functions in terms of the reduced functions. For the latter, we also characterize Fréchet differentiability. Finally, we obtain a generalization of Lidskiĭ's theorem on the spectrum of additive perturbations of Hermitian matrices to arbitrary spectral decomposition systems.

Variational Analysis in Spectral Decomposition Systems

TL;DR

The paper develops a unified variational analysis for spectral functions and spectral sets that depend only on eigenvalues or singular values by formalizing spectral decomposition systems. It derives comprehensive representations of Fréchet and limiting normal cones and subdifferentials, and establishes Fréchet differentiability transfer and Clarke subdifferentials in terms of invariant functions. A key contribution is a generalization of Lidskii's theorem that describes spectrum perturbations within arbitrary spectral decomposition systems. The framework subsumes eigenvalue and singular value settings, normal decomposition systems, and Euclidean Jordan algebras, enabling streamlined, geometry-based derivations of optimality conditions for matrix problems. Overall, the results unify disparate spectral analyses and open avenues for explicit optimality conditions and Lipschitz-like properties in spectral optimization problems.

Abstract

This work is concerned with variational analysis of so-called spectral functions and spectral sets of matrices that only depend on eigenvalues of the matrix. Based on our previous work [H. T. Bùi, M. N. Bùi, and C. Clason, Convex analysis in spectral decomposition systems, arXiv 2503.14981] on convex analysis of such functions, we consider the question in the abstract framework of spectral decomposition systems, which covers a wide range of previously studied settings, including eigenvalue decomposition of Hermitian matrices and singular value decomposition of rectangular matrices, and allows deriving new results in more general settings such as normal decomposition systems and signed singular value decompositions. The main results characterize Fréchet and limiting normal cones to spectral sets as well as Fréchet, limiting, and Clarke subdifferentials of spectral functions in terms of the reduced functions. For the latter, we also characterize Fréchet differentiability. Finally, we obtain a generalization of Lidskiĭ's theorem on the spectrum of additive perturbations of Hermitian matrices to arbitrary spectral decomposition systems.

Paper Structure

This paper contains 12 sections, 17 theorems, 118 equations.

Table of Contents

  1. Introduction
  2. Spectral decomposition systems
  3. Definitions, assumptions, and characterization
  4. Examples
  5. Fundamental properties
  6. Variational geometry of spectral sets
  7. Generalized subgradients of spectral functions
  8. Fréchet and limiting subdifferentials of spectral functions
  9. Fréchet differentiability of spectral functions
  10. Clarke subdifferentials of spectral functions
  11. A generalization of Lidskiı̆'s theorem
  12. Conclusion

Key Result

Proposition 2.4

Let $\mathfrak{H}$ be a Euclidean space, and let $(\mathcal{X},\mathsf{S},\gamma,(\Lambda_{\mathcalboondox{a}})_{\mathcalboondox{a}\in\mathcalboondox{A}})$ be a spectral decomposition system for $\mathfrak{H}$. Then a function $\Phi\colon\mathfrak{H}\to\intv{{-}\infty}{{+}\infty}$ is a spectral func and we call $\varphi$ the invariant function associated with $\Phi$.

Theorems & Definitions (55)

  • Definition 2.1: spectral decomposition system PartI
  • Definition 2.3: spectral function and spectral set
  • Proposition 2.4: PartI
  • Corollary 2.5: PartI
  • Example 2.7
  • proof
  • Example 2.8: normal decomposition system
  • proof
  • Example 2.9: Eaton triple
  • proof
  • ...and 45 more