Matrix Pencil Method

• Matrix Pencil Method is a linear algebra technique that represents parameterized matrices to extract spectral characteristics via generalized eigenvalue problems.
• It employs backward-stable algorithms, such as unitary QR decompositions, to compute the Kronecker canonical form and separate regular from singular parts.
• The method underpins applications in signal processing, parameter estimation, and statistical learning, offering robust super-resolution and interference mitigation.

A matrix pencil is a parametric family of matrices typically expressed as $A - \lambda B$, where $A$ and $B$ are fixed matrices over a field (usually $\mathbb{C}$ or $\mathbb{R}$), and $\lambda$ is a scalar parameter. Matrix pencils are central objects in mathematical analysis of linear differential-algebraic equations, numerical linear algebra, signal processing, parameter estimation, and machine learning. Their study encompasses both algebraic structure (canonical forms, invariants, equivalence) and algorithmic aspects (eigenvalue solvers, parameter extraction, signal reconstruction, and classification).

1. Algebraic Structure and Canonical Forms

The canonical analysis of a matrix pencil $(A,B)$ seeks to classify all pencils up to “strict equivalence” (or mixed equivalence): $(A,B)\sim(SA R, SB R)$ for invertible $S, R$. The Kronecker canonical form gives a complete invariant-based classification, decomposing any $m \times n$ pencil into a direct sum of:

• A regular part (Jordan blocks for non-defective eigenvalues; $\mathrm{det}(A - \lambda B)$ has full degree);
• Singular blocks encoding right and left minimal indices (correspond to violation of surjectivity or injectivity).

Explicitly, the Kronecker form presents $(A,B)$ as a block-diagonal matrix with components from the regular (invariant under similarity) and singular (canonical nilpotent or shift matrices) classes. The sizes and multiplicities of these blocks are the minimal indices (right $\alpha_k$, left $\beta_k^\pm$), which are intrinsic invariants of the pencil (Verdier, 2012, Klymchuk, 2018).

Regular pencils are those such that there exists $\lambda$ for which $\det(A-\lambda B)\not=0$. They admit a simple spectral characterization via the roots of $\det(A-\lambda B)$, generalizing the eigenvalues of a single matrix.

Canonical forms are computable using backward-stable algorithms—such as the unitary algorithm of Van Dooren, which computes the singular blocks using alternating QR and LQ decompositions, ensuring numerical reliability (Klymchuk, 2018).

2. The Matrix Pencil Method in Spectral Estimation and Signal Processing

The matrix pencil method (MPM) is a parametric modeling technique for recovering the parameters of a sum of complex exponentials or more general sparse structured signals. The classical MPM formulates the estimation as a generalized eigenvalue problem using Hankel or block-Hankel matrices constructed from the data:

Given $x[n] = \sum_{k=1}^K a_k z_k^n + w[n]$ ($w[n]$ noise), form two overlapping Hankel matrices $H_1$, $H_2$ from sliding windows of $x[n]$, and solve the pencil $H_1 v = \lambda H_2 v$. The generalized eigenvalues $\lambda_k$ are estimates of the system's poles $z_k$, and the amplitudes $a_k$ are recovered by a linear least-squares fit to the reconstructed Vandermonde matrix (Segman et al., 24 Feb 2025, wang et al., 2021, wang et al., 2022).

The method extends to multivariate, multi-dimensional, and multi-kernel data models, including sparse exponential sums in several variables. Multivariate matrix pencil extensions apply sequential SVD truncation, subspace projection, and joint diagonalization, and are robustified via techniques such as randomized mixing to reduce simultaneous diagonalization to eigenvalue computation for a single random matrix (Ehler et al., 2018, Bosner, 2020).

For data with missing values or gapped segmentations, the generalized matrix pencil method fuses local Hankel pencils from each segment by horizontal concatenation, enabling super-resolution parameter estimation across segmented or distributed arrays (wang et al., 2022).

3. Generalized Eigenvalue Problems and Algorithmic Workflows

The computational core of the matrix pencil method is solution of the generalized eigenvalue problem (EVP): $A v = \lambda B v~,$ where $A$ and $B$ come from data-dependent matrices (often block-Hankel, covariance, or commutator-derived matrices). This EVP admits a variety of fast and numerically robust solution strategies, including:

• SVD- or QR-based pre-projection onto dominant subspaces for noise-robust rank reduction (Pogorelyuk et al., 2018, Bosner, 2020, Segman et al., 24 Feb 2025);
• Inverse-free, highly-parallel randomized divide-and-conquer eigensolvers that achieve diagonalization of pencils in nearly matrix multiplication time and with backward-error guarantees (Demmel et al., 2023);
• Contour-integral based projection methods for interior or prescribed-region eigenvalues, extendable to nonsquare pencils via the Moore–Penrose pseudoinverse and subspace extraction using complex moments (Morikuni, 2020).

The method generalizes further to multiparameter pencils, as in the two-parameter problem $$A(\lambda_0, \lambda_1, \lambda_2) = \lambda_0 A_0 + \lambda_1 A_1 + \lambda_2 A_2$$, which is reduced to simultaneous eigenproblems via a Kronecker commutator-based symmetry and deflation procedure (Gungah et al., 2024).

For structured signals (e.g., sums over non-exponential or special functions), the multiscale MPM leverages generalized Hankel–Toeplitz matrices at multiple dilation and translation scales, enabling robust parameter and coefficient extraction for a wide class of signal models (cosine, sine, Chebyshev, Gaussian, etc.) (Cuyt et al., 2020). The principle remains: the critical nonlinear parameters are encoded in the spectrum of a structured pencil.

4. Statistical Learning and Differential Information Quantification

Matrix pencils with sample or population covariance matrices as constituents underpin several discriminant learning and statistical analysis frameworks. Given two covariance matrices $A$ (class 2) and $B$ (class 1), the pencil $A \tilde\psi = \mu B \tilde\psi$ arises in Fisher’s discriminant analysis, but can be interpreted more generally as a mechanism to extract "differential" information: the directions $\tilde\psi_k$ where variances differ most strongly between classes, with the generalized eigenvalues $\mu_k$ quantifying this ratio (Bhagat et al., 2020).

This approach enables binary or multi-class classification by constructing a feature space from principal axes of $B$ (whitened directions) and the pencil eigenvectors (differential directions), leading to remarkable empirical performance (e.g., near 99% accuracy for digit-pair MNIST classification using 5-NN in the pencil-B eigenbasis) (Bhagat et al., 2020). The method is fully algorithmic: compute $A$, $B$, whiten, extract pencil eigenvectors, project data, and train any off-the-shelf classifier.

The computational cost is dominated by the two $p \times p$ eigendecompositions of the covariances ($O(p^3)$ per class pair), but regularization or reduction to the dominant PCA subspace mitigates cost and handles rank-deficient scenarios.

5. Advanced Variants and Applications

Matrix pencil methodologies have been extended and adapted for a range of advanced signal processing and learning tasks:

• Robust model order selection and amplitude estimation: The Structure-Aware Matrix Pencil (SAMP) algorithm utilizes spectral properties of MP modes (columns of the left and right projected SVD factors) to reliably detect the number of active exponentials and efficiently estimate amplitudes, achieving near-theoretical Cramér-Rao lower bound performance even in low SNR and closely spaced frequency regimes (Segman et al., 24 Feb 2025).
• Super-resolution and sub-Nyquist sampling: CRT-based matrix pencil algorithms use congruent alias frequencies from multiple sub-Nyquist streams to reconstruct true frequencies via Chinese Remainder Theorem fusion, multiplying the effective resolution and sidestepping hardware limitations (Zhang et al., 2024).
• Multidimensional and multi-kernel parameter estimation: Methods such as the MDMP for massive MIMO channel prediction perform 3-D pencil construction in joint spatial, frequency, and time domains, provably achieving vanishing prediction error as antenna array size increases, and outperforming data-driven RNNs under high-mobility, large-delay conditions (Li et al., 2022).
• Generalized function fitting: Multiscale pencils enable robust, sample-efficient recovery for bases with shift–dilation invariance, and are effective for reconstructing sums of polynomials, special functions, or other structured kernels beyond the exponential (Cuyt et al., 2020).
• Interference mitigation and inpainting: Iterative pencil-based reconstruction algorithms can accurately recover missing signal segments in radar and gapped time series by fitting exponential or sinusoidal components to uncorrupted regions and blending across cut boundaries, outperforming AR-based and naive interpolation methods especially in low SNR and large-gap scenarios (wang et al., 2021, wang et al., 2022).
• Canonical form regularization: Algorithms for Kronecker form computation and regularization via unitary transformations guarantee numerical stability for large or ill-conditioned pencils, identifying singular structures and separating regular and singular blocks in a backward-stable manner (Klymchuk, 2018, Verdier, 2012).

6. Computational Considerations, Complexity, and Parallelism

The dominant computational costs in matrix pencil methods are matrix factorizations (SVD, eigendecomposition), typically $O(n^3)$ for $n \times n$ matrices, though parallel SVDs and block-power or Lanczos methods permit large-scale implementation. Structural regularization, block-diagonalization, and low-rank exploitation further reduce cost.

In multivariate cases, the formation and decomposition of large block-Hankel matrices can be efficiently parallelized: computational architectures such as hybrid CPU–GPU systems are particularly effective for matrix-matrix products and bulk SVD operations (Bosner, 2020). Randomized factorization and contour-based projection methods offer additional scalability and robustness to noise or ill-conditioning (Demmel et al., 2023, Morikuni, 2020).

Model order selection is commonly performed via singlular value thresholding, using AIC/MDL criteria or spectral separation of mode vectors (Segman et al., 24 Feb 2025).

7. Limitations and Theoretical Guarantees

Matrix pencil methods rely on several key assumptions:

• The target signal model (sum of exponentials or specified basis) must fit the data to within noise;
• The number of components $K$ must not exceed structural bounds set by data length and pencil dimensions;
• Adequate separation of system poles or kernel parameter values is required for robust decomposition and minimal parameter identifiability (Bosner, 2020, Cuyt et al., 2020);
• For classification/differential scenarios, covariance matrices must not be too ill-conditioned or low-rank; regularization or PCA reduction is necessary in high-dimensional, small-sample contexts (Bhagat et al., 2020).

Theoretical results include explicit stability bounds for parameter recovery under additive noise and tight non-asymptotic error controls, especially for super-resolution and multi-kernel unmixing (Chrétien et al., 2018).

Matrix pencil research continues to expand into high-dimensional, noisy, nonstationary, and multi-domain estimation, leveraging advances in parallel linear algebra, randomization, and low-rank modeling. The rigorous algebraic, numerical, and statistical foundations outlined above collectively enable scalable, interpretable, and theoretically grounded solutions in a wide range of modern data analysis and computational science applications.