Generalized Spectral Decomposition (GSD)

• Generalized Spectral Decomposition (GSD) is a unified framework that extends classical spectral methods to non-diagonalizable, non-normal, and higher-order operators.
• GSD employs coordinate-free decompositions for matrices, tensors, and infinite-dimensional operators, enabling robust analysis even with continuous or defective spectra.
• Its applications span signal processing, dynamical systems, and domain decomposition, providing systematic and computationally efficient spectral recovery techniques.

Generalized Spectral Decomposition (GSD) is a unifying framework extending classical spectral decomposition to a broad variety of linear and multilinear operators, matrices, tensors, categorical morphisms, and operator-theoretic objects. GSD provides coordinate-free decompositions even when diagonalization in the strict sense is not possible, generalizes to non-normal and defective operators, allows systematic handling of continuous spectra, and underlies advanced computational methods across applied mathematics, statistics, dynamical systems, signal processing, and category theory.

1. Foundational Notions of Spectral and Generalized Spectral Decomposition

Classical spectral decomposition expresses a diagonalizable matrix or normal operator as a sum or integral over projectors onto its eigenspaces weighted by eigenvalues. However, this does not extend to non-diagonalizable matrices (defective or non-normal), higher-order tensors, noncommutative or abstract categorical contexts, or operators with continuous spectrum. Generalized Spectral Decomposition addresses these limitations via several key constructions:

• Jordan and Projector-Based GSD: Any square matrix $A$ admits a generalized spectral decomposition over its primary subspaces (generalized eigenspaces) $\GG_i = \ker(A-\lambda_i I)^{m_i}$. The operator can be written

$$A = \sum_{i=1}^k \left( \lambda_i E_i + N_i \right)$$

where $E_i$ is the canonical projector (via Dunford's formula or polynomial in $A$) onto $\GG_i$, and $N_i = (A-\lambda_i I) E_i$ is a nilpotent part. Further decomposition extracts projectors $P_{ij}$ onto individual Jordan subspaces, yielding a unique, basis-independent decomposition for any $A$, diagonalizable or not (Deri et al., 2017).

• Categorial GSD: In semiadditive $\mathbf{C}\mathbf{Mon}$-categories (categories enriched in commutative monoids and admitting biproducts), an endomorphism $f:c\to c$ admits a GSD if $c$ splits into objects $x_1$, $x_2$ (generalizing eigenspaces) with arrows $\lambda_i:x_i\to x_i$ so that $$f = \kappa_1 \lambda_1 \rho_1 + \kappa_2 \lambda_2 \rho_2$$ (where $\kappa_i, \rho_i$ are canonical injections/projections). This construction is functorial and encompasses not only matrices but also relations, weighted graphs, and other algebraic and combinatorial data (Nishizawa et al., 24 Apr 2025).
• Infinite-Dimensional and Rigged Spaces: For operators with continuous spectrum, such as the Koopman operator in dynamical systems, GSD is formulated using rigged Hilbert spaces (Gelfand triplets) and resolution of the identity in terms of generalized eigenfunctions and spectral measures, encompassing both discrete and continuous parts (Colbrook et al., 2024).

2. GSD in Matrix, Tensor, and Array Analysis

Matrix Case

GSD provides a canonical spectral decomposition for arbitrary square matrices, subsuming the spectral theorem and Jordan decomposition: - The spectrum and block structure are extracted via minimal polynomial factorization. - Canonical projectors $E_i$ and $P_{ij}$ (onto generalized eigenspaces and Jordan blocks) are constructed as explicit algebraic expressions in $A$, or equivalently via contour integrals (Dunford). - The decomposition $A = \sum_{i,j} \lambda_i P_{ij} + N$ (with $N$ nilpotent) is unique and basis-independent, allowing unambiguous definition of generalized Fourier transforms (e.g., signal decomposition over graphs with defective shift operators) (Deri et al., 2017).

Tensor and Array Case

GSD extends beyond matrices, notably for real $I \times J \times 2$ arrays: - Given slices $Y_1, Y_2 \in \mathbb{R}^{I \times J}$, $Y$ admits an $R$-term GSD if there exist column-orthonormal $Q_a,Q_b$ and upper-triangular $R_1, R_2$ such that $Y_k = Q_a R_k Q_b^T$ ($k=1,2$). - The set $P_R(I,J,2)$ of GSD-representable arrays coincides with the closure $\overline{S}_R(I,J,2)$ of the set of tensors of CP-rank $\leq R$, ensuring the existence of best low-rank GSD approximations even when the best CP-rank-$R$ approximation does not exist (since $S_R$ is not closed). - Classification of arrays as interior/boundary/exterior points of $S_R$ corresponds to the real/degenerate/complex eigenstructure of a matrix pencil formed from $Y_1$, $Y_2$. - A Jacobi-type iterative solver efficiently computes the best GSD, avoiding the diverging behavior of non-closed rank loci (Stegeman, 2010).

3. Operator-Theoretic and Dynamical System GSD

Generalized spectral decomposition is critical for operator-theoretic analysis in infinite dimensions, particularly for non-normal or unitary operators with continuous spectrum:

• Koopman operators $\mathcal{K}$, acting on function spaces, generically possess continuous spectrum, precluding modal expansions in terms of eigenfunctions in $L^2$.
• Rigged DMD implements GSD by constructing a Gelfand triplet $\mathcal{S} \subset L^2 \subset \mathcal{S}^*$, enabling one to access generalized eigenfunctions $\psi \in \mathcal{S}^*$ and spectral density measures $\xi_k$. The expansion

$$g = \sum_{k\in \mathcal{I}} \int_{0}^{2\pi} \langle \psi_{\varphi,k}^* | g \rangle \; \psi_{\varphi,k} \, d\xi_{k}(\varphi)$$

captures both discrete and continuous spectral information.

• High-order kernel convolution with resolvent samples gives convergent approximations to generalized Koopman modes, providing practical algorithms with quantitative convergence guarantees in both discrete and continuous regimes (Colbrook et al., 2024).

4. GSD in Applied Problem Domains

System Identification

GSD underpins automated identification in ARX time series models:

• Parameters such as model order, delay, and coefficients are recovered from the generalized eigenstructure of stacked lag-covariance and noise-covariance matrices, $S_{Z_L} v = \lambda \Sigma_{e_L} v$.
• The scheme jointly infers system structure and parameters without a priori knowledge, robustly detecting true model order by counting eigenvalues close to unity and extracting ARX coefficients from corresponding eigenvectors.
• Iterative updating of the noise covariance via the Wiener–Khinchin relation completes a fully automated loop yielding consistent estimates even at low SNR, bypassing conventional try-and-test parametrization (Maurya et al., 2020).

Domain Decomposition and Preconditioning

GSD is integral to robust domain decomposition preconditioners in scientific computing:

• Localized GEVPs in overlapping subdomains define coarse basis functions (eigenvectors with eigenvalue $<\eta$), yielding scalable, mesh-robust, and coefficient-robust preconditioners for large-scale finite element and DG discretizations.
• Multilevel recursive application constructs a hierarchy of coarse spaces with guaranteed theoretical convergence bounds for symmetric positive definite problems, with strong empirical performance for diffusion, elasticity, and high-contrast problems (Bastian et al., 2021).

5. Categorical and Abstract GSD

The categorical generalization formalizes GSD in the context of semiadditive CMon-categories:

• Every hom-set is a commutative monoid, and objects admit all finite biproducts.
• The spectral decomposition is defined by splitting an endomorphism $f:c\to c$ into subobjects $(x_1,x_2)$ and arrows $(\lambda_1,\lambda_2)$ such that $f$ reconstructs via matrix-like sums of injections and projections, generalizing the direct sum of eigenspaces.
• This perspective unifies classical spectral theory, graph-theoretic splittings (e.g., equitable partitions), and decomposition in relational and fuzzy categories. Preservation of spectral decompositions under semiadditive functors establishes a robust categorical functoriality (Nishizawa et al., 24 Apr 2025).

6. Computational and Algorithmic Aspects

GSD-based algorithms exploit explicit or iterative construction of spectral projectors:

• For matrices: Spectral projectors and nilpotent parts are computed from minimal-polynomial factorizations, Jordan block analysis, and polynomial expressions or contour integral (Dunford) representations (Deri et al., 2017).
• For tensors: Jacobi-type iterative schemes alternate over orthogonal transformations and projection onto triangular slices, globally converging by virtue of the closedness of the GSD-representable set (Stegeman, 2010).
• For PDE/preconditioners: Localized eigenproblems yield coarse basis functions, with thresholding strategies controlling the coarse-space dimension and enabling multilevel additive Schwarz algorithms (Bastian et al., 2021).
• For Koopman operators: Data-driven resolvent sampling, kernel-based smoothing, and measure-preserving variants of EDMD ensure robust modal extraction in presence of nontrivial spectra (Colbrook et al., 2024).
• For ARX models: The QZ algorithm efficiently solves large-scale generalized eigenproblems, with spectral separation of unit modes guiding order estimation (Maurya et al., 2020).

7. Theoretical Significance and Broader Implications

Generalized spectral decomposition provides a universal toolkit for spectral analysis across linear algebra, tensor analysis, dynamical systems, statistical inference, operator theory, and category theory. In all these domains, GSD:

• Resolves the obstructions due to non-diagonalizability, non-normality, defective spectra, and lack of best low-rank approximations.
• Supplies algorithmic approaches that are robust to non-closed parameter loci, spectral degeneracy, and ill-conditioning.
• Underpins the generalization of physically and structurally meaningful decompositions—such as “harmonics” in graph signal processing—to contexts lacking orthogonal or symmetric structure.
• Bridges finite- and infinite-dimensional settings, unifying discrete, continuous, and categorical perspectives.

The versatility and analytic completeness of GSD ensure its continued expansion in theoretical research, computational mathematics, and broad interdisciplinary applications (Stegeman, 2010, Deri et al., 2017, Colbrook et al., 2024, Bastian et al., 2021, Maurya et al., 2020, Nishizawa et al., 24 Apr 2025).