Spectral decomposition of $(\star, ε)$-palindromic matrix polynomial and its applications

Abstract: This paper provides the spectral decomposition of $(\star,ε)$-palindromic quadratic matrix polynomial $P(λ)$ by a standard pair and a parameter matrix. When $J$ is assumed to be a block diagonal matrix, the parameter matrix $Γ$ has a special structure. And then the spectral decomposition is applied to solve the inverse eigenvalue problem and the eigenvalue embedding problem with no spill-over.

• The paper introduces a unified spectral decomposition for (⋆, ε)-palindromic matrix polynomials, establishing necessary and sufficient conditions for the standard pair.
• It formulates a structured parameter matrix Γ that enables the explicit reconstruction of coefficient matrices A and Q with high numerical precision.
• The framework supports robust solutions for quadratic inverse eigenvalue and eigenvalue embedding problems, ensuring no spill-over in the preserved spectrum.

Spectral Decomposition of $(\star, \epsilon)$-Palindromic Matrix Polynomials

Introduction and Problem Formulation

This paper develops a unified spectral decomposition framework for $(\star, \epsilon)$-palindromic quadratic matrix polynomials:

$P(\lambda)=\lambda^2A+\lambda Q+\epsilon A^{\star}$

where $\star \in \{H, T\}$ indicates Hermitian or transposed conjugate, $\epsilon \in \{1, -1\}$, and $Q^{\star}=\epsilon Q$. Such polynomials have eigenvalues occurring in reciprocal and conjugate pairs, and subsume $T$-palindromic, $T$-anti-palindromic, $H$-palindromic, and $H$-anti-palindromic forms. These classes arise in the modeling and numerical analysis of advanced dynamical systems—e.g., vibration analysis and vibroacoustic systems—where symmetry and structure preservation are critical due to strong physical or engineering constraints.

The core objective is to derive the exact spectral decomposition for this general class, explicitly characterizing the roles of a standard pair $(\star, \epsilon)$0—with $(\star, \epsilon)$1 block-diagonal and $(\star, \epsilon)$2 composed of generalized eigenvectors—and a parameter matrix $(\star, \epsilon)$3 encoding algebraic structure. The decomposition is systematically leveraged for two fundamental problems: the inverse eigenvalue problem (IEP), and the eigenvalue embedding problem (EEP) with no spill-over.

Main Results: Spectral Decomposition Theory

Unified Spectral Decomposition

The principal contribution is a necessary and sufficient condition for $(\star, \epsilon)$4 to be a standard pair of $(\star, \epsilon)$5, expressed as:

$(\star, \epsilon)$6

together with $(\star, \epsilon)$7 nonsingular, and the introduction of the structure set:

$(\star, \epsilon)$8

A unique parameter matrix $(\star, \epsilon)$9 is shown to satisfy:

$P(\lambda)=\lambda^2A+\lambda Q+\epsilon A^{\star}$0

and, crucially, the coefficient matrices can be reconstructed as:

$P(\lambda)=\lambda^2A+\lambda Q+\epsilon A^{\star}$1

This formulation explicitly encodes the problem structure and constraints in terms of parameter spaces and conventional Jordan-theoretic invariants.

Structure of $P(\lambda)=\lambda^2A+\lambda Q+\epsilon A^{\star}$2

For block-diagonal $P(\lambda)=\lambda^2A+\lambda Q+\epsilon A^{\star}$3 (Jordan canonical; potentially real-represented), $P(\lambda)=\lambda^2A+\lambda Q+\epsilon A^{\star}$4 decomposes as:

$P(\lambda)=\lambda^2A+\lambda Q+\epsilon A^{\star}$5

where each block is determined by the spectral symmetries of individual eigenvalues and their algebraic multiplicities.

Distinct constructions arise for different palindromic classes. For instance, $P(\lambda)=\lambda^2A+\lambda Q+\epsilon A^{\star}$6- (anti-)palindromic cases with real fields lead to block structures based on $P(\lambda)=\lambda^2A+\lambda Q+\epsilon A^{\star}$7, while $P(\lambda)=\lambda^2A+\lambda Q+\epsilon A^{\star}$8-palindromic cases in the complex field require more nuanced handling for unit-modulus eigenvalues.

Special attention is given to the semi-simple case, where all eigenvalues are simple or have geometric multiplicity equal to their algebraic multiplicity. Here, $P(\lambda)=\lambda^2A+\lambda Q+\epsilon A^{\star}$9 can be reduced to direct sums of identity or signature matrices up to congruence, significantly simplifying parametrization and computation.

Applications: Inverse Problems and Eigenvalue Embedding

Quadratic Inverse Eigenvalue Problem (QIEP)

Given a target eigenstructure $\star \in \{H, T\}$0, the framework characterizes all $\star \in \{H, T\}$1-palindromic polynomials admitting it via parameter selection $\star \in \{H, T\}$2:

$\star \in \{H, T\}$3

The existence of a nonsingular $\star \in \{H, T\}$4 of specified form ensuring $\star \in \{H, T\}$5 and $\star \in \{H, T\}$6 invertible suffices for solution; formulae for $\star \in \{H, T\}$7 and $\star \in \{H, T\}$8 follow directly. Strong numerical demonstrations are provided with spectral accuracy ($\star \in \{H, T\}$9-residuals $\epsilon \in \{1, -1\}$0) on both $\epsilon \in \{1, -1\}$1- and $\epsilon \in \{1, -1\}$2-palindromic cases. The flexibility and constructiveness of this approach outperforms prior ad hoc or structure-agnostic techniques.

Eigenvalue Embedding Problem (EEP) with No Spill-Over

This problem, crucial to model updating/control, seeks to embed/reassign a subset of eigenvalues ($\epsilon \in \{1, -1\}$3) while maintaining the remaining spectrum and preserving structure. The decomposition yields an explicit algorithmic construction: for any prescribed substitution $\epsilon \in \{1, -1\}$4, and blocks $\epsilon \in \{1, -1\}$5 with

$\epsilon \in \{1, -1\}$6

the updated polynomial is computed via:

$\epsilon \in \{1, -1\}$7

with $\epsilon \in \{1, -1\}$8 analogous. This approach naturally enforces no spill-over: the untouched eigenpairs are left unchanged—a property critical in model updating and robust control applications.

Multiple numerical examples reveal that prescribed eigenvalue replacements are realized up to machine precision, with no alteration to the rest of the spectrum, even for modulus-one and clustered eigenvalues. Notably, modulus-one eigenvalues in $\epsilon \in \{1, -1\}$9-anti-palindromic problems can only be replaced by other modulus-one values, reflecting deeper algebraic invariants.

Implications and Future Work

The presented unified decomposition not only solves longstanding problems in the characterization and synthesis of structured matrix polynomials but also provides a foundation for high-fidelity numerical algorithms in inverse eigenvalue problems and robust control. The explicit parametric description of admissible coefficient matrices and the detailed block structure theory for $Q^{\star}=\epsilon Q$0 enable efficient, structure-preserving computation at scale.

Theoretically, this work clarifies the symmetry-induced restrictions on spectrum assignability (e.g., reducibility of modulus-one eigenvalue embeddings in $Q^{\star}=\epsilon Q$1-anti-palindromic forms), and further opens up the possibility for robust optimization in the presence of spectral symmetries.

Potential future developments include fast numerical algorithms for large-scale semi-simple cases using the block decomposition of $Q^{\star}=\epsilon Q$2, extension to higher-order palindromic polynomials, and the exploitation of the decomposition for advanced system identification and structured feedback design in control engineering.

Conclusion

This paper establishes the spectral decomposition for $Q^{\star}=\epsilon Q$3-palindromic quadratic matrix polynomials in a unified algebraic framework, anchored by the standard pair and structured parameter $Q^{\star}=\epsilon Q$4. The practical impact is illustrated through explicit, numerically robust algorithms for both the inverse eigenvalue problem and no spill-over eigenvalue embedding, applicable to key engineering and numerical analysis domains. The theoretical insights into admissible structures and parameterizations constitute a comprehensive foundation for the analysis and synthesis of structured polynomial eigenvalue problems in applied mathematics.