Spectral decomposition of $(\star,ε_1,ε_2)$-structured matrix polynomials with arbitrary degree and its applications

Abstract: This paper provides the spectral decompositions of $(\star,ε_1,ε_2)$-structured matrix polynomials $P(λ)$ in the unified form by a standard pair and parameter matrix. Using the recursive relationship between the coefficient matrices of $P(λ)$, equivalent expressions of these coefficient matrices are provided. 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 framework that parametrically constructs structured matrix polynomials using standard pairs and a novel parameter matrix.
• It establishes explicit, recursive formulations for all coefficient matrices under diverse symmetry constraints in both regular and singular cases.
• It applies the theory to resolve inverse eigenvalue and no-spillover embedding problems, providing constructive algorithms for model updating in vibration and control systems.

Spectral Decomposition of $(\star,\epsilon_1,\epsilon_2)$-Structured Matrix Polynomials: Unified Theory and Applications

Overview

This paper develops a comprehensive theory for the spectral decomposition of structured matrix polynomials of arbitrary degree, where the structure is defined by $(\star, \epsilon_1, \epsilon_2)$, encompassing a full taxonomy: $T$-symmetric, $T$-skew-symmetric, $T$-even, $T$-odd, $H$-Hermitian, $H$-skew-Hermitian, $H$-even, and $H$-odd. The framework unifies these cases via a parameterization involving standard pairs and a newly clarified parametric matrix with precise structure determined by the eigenvalue type and the associated symmetry or skew-structure. Applications span the inverse eigenvalue problem (IEP) and the no-spillover eigenvalue embedding problem (EEP), both of which are central in vibration analysis, structure-preserving model updating, and high-order linear control systems.

Unified Spectral Decomposition

The central object of study is a matrix polynomial $(\star, \epsilon_1, \epsilon_2)$0 with $(\star, \epsilon_1, \epsilon_2)$1 and structure constraints: $(\star, \epsilon_1, \epsilon_2)$2 (even $(\star, \epsilon_1, \epsilon_2)$3) or $(\star, \epsilon_1, \epsilon_2)$4 (odd $(\star, \epsilon_1, \epsilon_2)$5), with $(\star, \epsilon_1, \epsilon_2)$6, $(\star, \epsilon_1, \epsilon_2)$7. This generalizes and unifies a variety of previously studied classes. The eigenstructure is determined by the values of $(\star, \epsilon_1, \epsilon_2)$8, with configurations leading to real, complex, Hermitian, skew-Hermitian, even, odd, or palindromic eigenpair symmetries, as enumerated in the literature.

The decomposition is given in terms of a standard pair $(\star, \epsilon_1, \epsilon_2)$9, where $T$0 has block-Jordan structure reflecting the algebraic and geometric multiplicities and eigenvalue configuration, and $T$1 is the corresponding matrix of (generalized) eigenvectors. The innovation of this work is to provide a parametric formula for all coefficient matrices $T$2 using $T$3 and a single parameter matrix $T$4, whose block structure and symmetry are explicitly characterized. The main result (Theorem 1) shows that for regular polynomials (nonsingular leading coefficient), all structural classes admit such a decomposition, with coefficient matrices given recursively via $T$5 under explicit orthogonality-type constraints. The approach also extends to the singular case (e.g., polynomials with infinite eigenvalues) via a duality argument.

Structural Characterization of Parameter Matrices

The parameter matrix $T$6 must respect induced structure conditions (blockwise symmetry, Hankel or skew-Hankel, Hermitian/skew-Hermitian, etc.) dictated by the structure class of $T$7 and the Jordan canonical form of $T$8. Lemma 3 and Theorem 4 provide detailed blockwise descriptions, extending and generalizing results known for second-order or unstructured polynomials. The block diagonal form of $T$9 aligns with the block structure of $T$0, and every block is further restricted (e.g., to upper triangular Hankel or skew-Hankel blocks). This enables direct construction of $T$1 with prescribed structural constraints in practical applications.

Applications to Inverse Eigenvalue and Eigenvalue Embedding Problems

Strong attention is paid to concrete algorithmic applications:

Inverse Eigenvalue Problem (IEP)

Given a desired set of (structured) eigenpairs, the decomposition framework parametrically generates all coefficient matrices $T$2 compatible with the assigned spectral data and structural constraints. This closes an important gap versus prior work (Gohberg–Lancaster–Rodman theory) which did not consider structural preservation, and enables explicit parametric solutions for arbitrary degree $T$3.

Eigenvalue Embedding Problem (EEP) with No Spill-over

The most technically significant application is to model updating, particularly in vibration engineering and control, where selected eigenvalues (and associated eigenspaces) must be replaced or embedded without altering the remainder (no spill-over phenomenon) and while fully preserving system structure. The paper rigorously characterizes when such an update is feasible and presents constructive algorithms for both quadratic and higher-degree structured polynomials. For $T$4 (quadratic case), the solution is always possible, and explicit formulas for the embedding update are provided. For $T$5, the analysis identifies sufficient solvability conditions, and for $T$6-symmetric quartic polynomials ($T$7), explicit procedures are presented.

The practical component is supported by algorithmic recipes and concrete numerical examples, demonstrating the effectiveness of the proposed spectral decomposition framework for structured model updating. In all cases, the preserved or embedded eigenpairs satisfy prescribed structure-induced multiplicity and symmetry constraints (e.g., quadruple occurrence in $T$8-even/odd cases).

Numerical and Algorithmic Considerations

The recursive expressions for $T$9 are reformulated to minimize the number of terms requiring explicit use of the standard pair, improving numerical and symbolic tractability. The algorithms provided exploit congruence and orthogonal/unitary transformations to generate parametrizations for the free matrices in the embedding problem, particularly for handling blocks associated with real, complex, and pure imaginary eigenvalues. Several cases—such as the existence of solutions for $T$0 (guaranteed solvability) and sufficient conditions under which higher-order cases are solvable—are discussed with clarity.

Implications and Future Directions

Practically, this unified spectral decomposition theory facilitates robust and structure-preserving model updating for high-order linear systems, enabling explicit controller/observer design and system identification in applications such as mechanical vibration analysis, robotics, and flexible structure control. Theoretically, the parametrization clarifies the geometry of the solution sets for both IEP and EEP in the structured setting, bridging a longstanding gap between unstructured and structured polynomial eigenproblems.

Notably, the explicit block-structured nature of the parameter matrix $T$1 and its role in constructing all possible polynomials with prescribed spectral and structural properties offers a platform for further developments, such as statistical sampling in parametric uncertainty quantification, or the study of structure-preserving perturbations and robust control synthesis.

A key open challenge highlighted is the characterization of solutions to system equations (e.g., (6) and (gzs-10)) for arbitrary $T$2 and arbitrary degree $T$3, especially in the context of general structured polynomials. This points to avenues for further research in both symbolic computation and numerical algorithms for large-scale structured eigenproblems.

Conclusion

The paper accomplishes a unified spectral decomposition framework for all classical structure classes of matrix polynomials of arbitrary degree, fully characterizes the associated parametric families, and demonstrates explicit solutions to both inverse and embedding eigenvalue problems while rigorously preserving structural properties. This work generalizes and subsumes many prior special cases, advancing both the theoretical understanding and the computational practice of structured polynomial eigenproblems (2606.07176).