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Matrix pencil

A matrix pencil is a pair of complex matrices A and B of the same dimensions, typically square, with the pencil defined as the linear matrix polynomial A - \lambda B, where \lambda is a scalar parameter.^[1] This structure generalizes the standard eigenvalue problem A\mathbf{v} = \lambda \mathbf{v} to the generalized form A\mathbf{v} = \lambda B\mathbf{v}, where the eigenvalues are the values of \lambda for which \det(A - \lambda B) = 0.^[2] A pencil is termed regular if its determinant is not identically zero, ensuring a well-defined characteristic polynomial of degree equal to the matrix size, while singular or irregular pencils lack this property and exhibit more complex behavior.^[1] The theory of matrix pencils encompasses their classification via the Kronecker canonical form, which decomposes a pencil into a direct sum of blocks representing eigenvalues, Jordan chains, and singular structures like right and left null spaces, providing a complete invariant under strict equivalence transformations P(A - \lambda B)Q for invertible P and Q.^[3] For regular pencils, this reduces to the generalized Jordan form, while singular cases include rectangular blocks that capture infinite eigenvalues and minimal indices.^[4] Numerical methods, such as the QZ algorithm, compute this form by simultaneously triangularizing A and B via unitary transformations, enabling stable eigenvalue extraction even for ill-conditioned problems.^[1] Matrix pencils arise prominently in applications across engineering and applied mathematics, including the analysis of descriptor systems in control theory modeled as E\dot{x} = Ax + Bu where E may be singular, vibration problems in structural dynamics, and parameter estimation in signal processing via methods like the matrix pencil technique for exponential signal decomposition.^[5] They also feature in optimization, such as semidefinite programming with Hermitian pencils, and in solving polynomial eigenvalue problems through linearization, where higher-degree polynomials are reduced to equivalent pencils while preserving spectral properties.^[6] Perturbation theory for pencils addresses sensitivity of eigenvalues and deflating subspaces, crucial for robust numerical implementations.^[3]

Basic Concepts

Definition

A matrix pencil is defined as a pair of matrices A and B of the same dimensions m \times n over a field, typically the complex numbers, denoted as the pencil (A, B) or expressed as the matrix-valued polynomial A - \lambda B, where \lambda is an indeterminate scalar variable.^[7] The concept originated in the late 19th century through the work of Leopold Kronecker on canonical forms for pairs of matrices, as detailed in his 1890 paper on the algebraic reduction of sheaves of bilinear forms.^[8] Alternative notations appear in different fields; for instance, control theory often uses sA - B, while eigenvalue problems may employ \lambda B - A, but these conventions do not alter the fundamental structure of the pencil.^[9] For square pencils, the pencil is linear, having degree 1, and corresponds to the generalized eigenvalue problem \det(A - \lambda B) = 0.^[10] For illustration, consider the $2 \times 2 pencil with

A = \begin{pmatrix} 1 & 0 \\ 0 & 2 \end{pmatrix}, \quad B = \begin{pmatrix} 0 & 1 \\ 1 & 0 \end{pmatrix},

yielding

A - \lambda B = \begin{pmatrix} 1 & -\lambda \\ -\lambda & 2 \end{pmatrix}.

This example demonstrates the polynomial form without specifying eigenvalues or further properties.^[7]

Regular and Singular Pencils

A square matrix pencil A - \lambda B, where A and B are square matrices of the same order n, is defined as regular if the determinant \det(A - \lambda B) is not identically zero as a polynomial in \lambda.^[11] Otherwise, the pencil is singular.^[11] This classification is fundamental because it determines the solvability and structure of the associated generalized eigenvalue problem A x = \lambda B x.^[11] For a regular pencil, there are exactly n eigenvalues, counting algebraic multiplicities, which may be finite or infinite.^[11] Singular pencils, in contrast, possess fewer than n eigenvalues and exhibit structural defects, such as dependencies in rows or columns that persist across all \lambda.^[11] These defects manifest in the pencil's Kronecker canonical form through parameters known as row and column minimal indices, which quantify the degrees of the polynomial bases for the right and left null spaces.^[11] Infinite eigenvalues arise when B is singular and are formally defined by considering the reciprocal eigenvalues of the reversed pencil B - \mu A, where \mu = 1/\lambda, or through a homogenization approach viewing the pencil as the homogeneous equation \mu A - \nu B = 0 in projective space with coordinates [\mu : \nu].^[11] In the projective interpretation, an infinite eigenvalue corresponds to the point at infinity [0 : 1].^[11] Consider a 2×2 example of a regular pencil: let A = \begin{pmatrix} 1 & 0 \\ 0 & 2 \end{pmatrix} and B = I_2, yielding \det(A - \lambda B) = (1 - \lambda)(2 - \lambda), with finite eigenvalues 1 and 2.^[11] In contrast, a singular pencil such as A = \begin{pmatrix} 1 & 0 \\ 0 & 0 \end{pmatrix} and B = \begin{pmatrix} 0 & 1 \\ 0 & 0 \end{pmatrix} has \det(A - \lambda B) = 0 identically, as the second row is always zero, indicating linear dependence for all \lambda.^[11] For singular pencils, the dimension of the kernel of A - \lambda B is at least 1 for every \lambda \in \mathbb{C} \cup \{\infty\}, reflecting the constant rank deficiency.^[11] This varying kernel structure, beyond simple eigenvalue multiplicities, introduces the minimal indices that characterize the pencil's null space polynomials.^[11]

Properties

Algebraic Properties

A matrix pencil A - \lambda B, where A and B are m \times n complex matrices, is subject to the equivalence relation known as strict equivalence. Two pencils A - \lambda B and A' - \lambda B' are strictly equivalent if there exist invertible matrices P \in \mathbb{C}^{m \times m} and Q \in \mathbb{C}^{n \times n} such that A' = P A Q and B' = P B Q.^[12] This transformation preserves the generalized eigenvalues of the pencil, as the roots of \det(A - \lambda B) (when defined) remain unchanged up to the action of P and Q.^[12] Under strict equivalence, several algebraic invariants characterize the structure of the pencil. For regular square pencils, the Segre characteristics provide a complete set of invariants; these are the lists of sizes of the Jordan blocks corresponding to each distinct finite generalized eigenvalue, ordered non-increasingly for each eigenvalue.^[12] For singular pencils, additional Kronecker invariants are required, including the column minimal indices (the degrees of a minimal basis for the null space of the pencil over the rational function field) and row minimal indices (analogously for the left null space).^[12] These invariants together determine the equivalence class uniquely.^[12] The normal rank of a matrix pencil A - \lambda B is defined as the maximum value of \operatorname{rank}(A - \lambda B) over all \lambda \in \mathbb{C}, or equivalently, the rank of the pencil when viewed as a matrix over the field of rational functions \mathbb{C}(\lambda).^[13] For an n \times n square pencil, the normal rank equals n if and only if the pencil is regular, meaning \det(A - \lambda B) is not identically zero.^[12] Determinantal divisors further describe the algebraic structure of the pencil. For each k = 1, \dots, r where r is the normal rank, the k-th determinantal divisor \Delta_k(\lambda) is the greatest common divisor of all k \times k minors of A - \lambda B, treated as polynomials in \lambda.^[12] The invariant factors of the pencil are then obtained as ratios of consecutive determinantal divisors: d_k(\lambda) = \Delta_k(\lambda) / \Delta_{k-1}(\lambda) (with \Delta_0(\lambda) = 1), providing a diagonal form under equivalence over the polynomial ring.^[12] As an illustrative example, consider the $2 \times 2 pencil

A - \lambda B = \begin{pmatrix} \lambda + 1 & 2 \\ 3 & \lambda + 4 \end{pmatrix}.

Through strict equivalence transformations, this can be brought to the form \operatorname{diag}(1, (\lambda + 1)(\lambda + 4) - 6), revealing the invariant factors d_1(\lambda) = 1 and d_2(\lambda) = \lambda^2 + 5\lambda - 2, which are preserved under equivalence.^[12]

Spectral Properties

The spectral properties of a matrix pencil A - \lambda B, where A and B are n \times n complex matrices, center on its eigenvalues and associated structures, assuming the pencil is regular (i.e., \det(A - \lambda B) is not identically zero).^[14] The generalized eigenvalues are the complex numbers \lambda satisfying \det(A - \lambda B) = 0, which are the roots of the characteristic polynomial p(\lambda) = \det(A - \lambda B).^[14] For a regular pencil, this polynomial has degree exactly n, and the algebraic multiplicity of an eigenvalue \lambda is the multiplicity of \lambda as a root of p(\lambda).^[15] Infinite eigenvalues arise when \det(B) = 0, leading to a drop in the degree of p(\lambda) below n; the multiplicity of the infinite eigenvalue is then n - \deg(p(\lambda)).^[14] Equivalently, an infinite eigenpair (x, \infty) satisfies Bx = 0 with x \neq 0, provided the pencil remains regular.^[15] The geometric multiplicity of a finite eigenvalue \lambda is the dimension of the eigenspace \ker(A - \lambda B), which equals the number of linearly independent eigenvectors associated with \lambda.^[14] For each eigenvalue \lambda (finite or infinite), the structure is captured by Jordan chains of generalized eigenvectors. For a finite eigenvalue \lambda, a Jordan chain of length k is a sequence of vectors v_1, \dots, v_k satisfying (A - \lambda B) v_1 = 0 and (A - \lambda B) v_j = B v_{j-1} for j = 2, \dots, k, with v_j \neq 0. For an infinite eigenvalue, the chains are defined analogously using the reciprocal pencil B - \mu A (with \mu = 1/\lambda = 0), where a chain w_1, \dots, w_k satisfies B w_1 = 0 and A w_j = B w_{j-1} for j = 2, \dots, k.^[14] The lengths of these chains are given by the Segre characteristics, which describe the sizes of the Jordan blocks in the Weierstrass canonical form and determine the ascent of the eigenvalue.^[16] The number of such chains equals the geometric multiplicity. Consider the regular 3×3 pencil with

A = \begin{pmatrix} 9 & 8 & 7 \\ 6 & 5 & 4 \\ 3 & 2 & 1 \end{pmatrix}, \quad B = \begin{pmatrix} 1 & 3 & 2 \\ 4 & 6 & 5 \\ 7 & 9 & 8 \end{pmatrix}.

The characteristic polynomial is p(\lambda) = \det(A - \lambda B), with roots (generalized eigenvalues) approximately \lambda_1 \approx 1.8984, \lambda_2 = -1, \lambda_3 \approx -0.0807, each of algebraic multiplicity 1.^[17] Since the eigenvalues are distinct, the geometric multiplicity is 1 for each, and the Jordan chains are trivial (length 1), consisting solely of the corresponding eigenvectors. No infinite eigenvalues occur, as \det(B) \neq 0 and \deg(p(\lambda)) = 3.^[17]

Canonical Forms

Jordan Canonical Form for Regular Pencils

A regular matrix pencil A - \lambda B of size n \times n is strictly equivalent to a unique (up to permutation of blocks) Jordan canonical form, consisting of a block-diagonal structure that reveals the Jordan chains associated with its finite eigenvalues and the infinite eigenvalue. This form exists because the pencil is regular, meaning \det(A - \lambda B) \not\equiv 0, allowing a decomposition into invariant subspaces corresponding to the generalized eigenspaces. The uniqueness follows from the invariance of the sizes and number of Jordan blocks, known as the Segre characteristics, under strict equivalence transformations.^[18] The structure comprises standard Jordan blocks for each finite eigenvalue \mu and modified blocks for the infinite eigenvalue. For a finite eigenvalue \mu, each block of size k \times k is given by J_k(\mu) - \lambda I_k, where J_k(\mu) is the Jordan block with \mu on the diagonal and 1's on the superdiagonal; explicitly,

J_k(\mu) = \begin{pmatrix} \mu & 1 & & \\ & \mu & \ddots & \\ & & \ddots & 1 \\ & & & \mu \end{pmatrix}.

This results in the shifted structure (\mu - \lambda) I_k + N_k, where N_k is the nilpotent Jordan block. For the infinite eigenvalue, the blocks are of the form I_l - \lambda N_l, where N_l is the l \times l nilpotent Jordan block (0's on the diagonal, 1's on the superdiagonal), capturing the chains at infinity via equivalence to the reversed pencil B - \mu A with \mu = 1/\lambda. These infinite blocks arise from the nilpotent part in the Weierstrass decomposition, ensuring the full pencil is block-diagonal.^[18]^[19] The transformation to this form is achieved by finding nonsingular matrices P and Q such that

P(A - \lambda B)Q = \bigoplus_i (J_{k_i}(\mu_i) - \lambda I_{k_i}) \oplus \bigoplus_j (I_{l_j} - \lambda N_{l_j}),

where the direct sum assembles all finite and infinite blocks to match the original dimension n. When B = I_n, the pencil simplifies to A - \lambda I_n, and the form reduces precisely to the classical Jordan canonical form of the matrix A, with no infinite blocks.^[18]^[19] The Jordan canonical form for regular pencils is typically computed via the generalized Schur decomposition, produced by the QZ algorithm, which triangularizes the pencil while preserving its equivalence class and from which the block structure can be extracted. As an illustrative example, consider a $4 \times 4 regular pencil with one Jordan chain of length 2 for the finite eigenvalue \mu = 1 and one of length 2 for the infinite eigenvalue. The canonical form is block-diagonal:

\begin{pmatrix} 1 - \lambda & 1 & 0 & 0 \\ 0 & 1 - \lambda & 0 & 0 \\ 0 & 0 & 1 & -\lambda \\ 0 & 0 & 0 & 1 \end{pmatrix},

where the top-left $2 \times 2 block corresponds to the finite eigenvalue and the bottom-right $2 \times 2 block (I_2 - \lambda N_2 with N_2 superdiagonal) to the infinite eigenvalue. This decomposition highlights the algebraic multiplicities and defect structures without requiring the explicit P and Q.^[18]

Kronecker Canonical Form for Singular Pencils

The Kronecker canonical form generalizes the Jordan canonical form to singular matrix pencils, providing a complete decomposition that accounts for both the regular and irregular structures under strict equivalence. Developed by Leopold Kronecker in 1890, it extends earlier work on regular pencils, such as the Weierstrass form, by incorporating blocks that capture the singularities arising from rank deficiencies.^[20]^[7] For a singular pencil A - \lambda B \in \mathbb{C}^{m \times n}, there exist nonsingular matrices P \in \mathbb{C}^{m \times m} and Q \in \mathbb{C}^{n \times n} such that

P (A - \lambda B) Q = \bigoplus_i (J_i(\mu_i) - \lambda I_i) \oplus \bigoplus_j (\lambda N_j - I_j) \oplus \bigoplus_k L_k \oplus \bigoplus_l L_l^T,

where the direct sum consists of Jordan blocks J_i(\mu_i) for finite eigenvalues \mu_i, nilpotent Jordan blocks N_j for the eigenvalue at infinity (reversing the roles of A and B), right singular blocks L_k of size k \times (k+1), and left singular blocks L_l^T of size (l+1) \times l. The blocks L_k are defined as the k × (k+1) pencils with 1's on the superdiagonal (in the A part) and -λ on the diagonal (from -λ times diagonal in B), reflecting the minimal degree rational vector bases in the right nullspace; similarly, L_l^T handle the left nullspace. These singular blocks appear precisely when the pencil is singular, i.e., when \det(A - \lambda B) \equiv 0.^[7]^[21] The column minimal indices are the integers k associated with each L_k block, representing the degrees of a basis for the rational right nullspace of the pencil. Analogously, the row minimal indices l correspond to the L_l^T blocks for the left nullspace. These indices directly relate to the pencil's defect, defined as d = (m + n) - 2r, where r is the normal rank (the maximum rank of A - \lambda B over \lambda); specifically, the sum of the column minimal indices plus the sizes of the regular blocks equals n - r, and similarly for rows. The minimal indices, together with the degrees of the elementary divisors from the regular parts, form the complete set of invariants under strict equivalence.^[7]^[21] This canonical form is unique up to permutation of the blocks of the same type, thereby classifying all singular pencils and distinguishing their structural properties. For non-square pencils, the rectangular nature is preserved through the differing dimensions of the singular blocks, ensuring the total row and column counts match m and n. Strict equivalence applies via the nonsingular P and Q, even for rectangular cases, as long as the transformations maintain the appropriate sizes.^[7]^[22] As an illustrative example, consider a singular $3 \times 4 pencil with one column minimal index of 0 (corresponding to an L_0 block, a $0 \times 1 structure indicating a free column with no row constraint) and a regular Jordan part consisting of a $3 \times 3 block for a finite eigenvalue. The resulting Kronecker form is a block-diagonal arrangement where the regular $3 \times 3 subpencil handles the eigenvalue structure, while the L_0 accounts for the extra column, demonstrating the mixed regular-singular composition that accommodates the dimension mismatch and rank defect.^[7]

Special Classes

Pencils Generated by Commuting Matrices

A matrix pencil generated by commuting matrices consists of a pair (A, B) where A and B are square matrices satisfying AB = BA. Over an algebraically closed field such as the complex numbers, such pairs are simultaneously triangularizable, meaning there exists an invertible matrix P such that both P^{-1}AP and P^{-1}BP are upper triangular.^[23] This property follows from the classical theorem on commuting matrices, which ensures a common chain of invariant subspaces.^[24] For a regular pencil (A, B) where A and B commute, the matrices share a common eigenbasis if both are diagonalizable. In this case, the generalized eigenvalues of the pencil \lambda B - A are the ratios \lambda_i = \alpha_i / \beta_i, where \alpha_i and \beta_i are the corresponding joint eigenvalues of A and B for the common eigenvector v_i, provided \beta_i \neq 0. If some \beta_i = 0, the pencil may have infinite eigenvalues or require careful handling of the kernel.^[25] This common eigenbasis implies that the pencil admits a diagonal form in the generalized sense, facilitating the computation of its spectrum as these ratios.^[25] Due to commutativity, the canonical form of such a pencil reduces to a block-diagonal structure simpler than the general case. If both A and B are diagonalizable, the pencil \lambda B - A is diagonalizable via the common similarity transformation, yielding a diagonal matrix with entries \lambda \beta_i - \alpha_i. In the broader Kronecker canonical form, commutativity imposes restrictions on the block types, often resulting in a direct sum of Jordan blocks without singular (Kronecker) blocks when the pencil is regular.^[26] Consider the example of two commuting diagonal matrices A = \operatorname{diag}(1, 2, 3) and B = \operatorname{diag}(4, 5, 6), which trivially share the standard basis as their common eigenbasis. The pencil \lambda B - A has generalized eigenvalues \lambda_1 = 1/4, \lambda_2 = 2/5, and \lambda_3 = 3/6 = 1/2, confirming the ratio property. This pencil is already in diagonal form, illustrating the full diagonalizability.^[25] The spectrum of the pencil directly corresponds to the ratios of the joint eigenvalues of the commuting pair (A, B). These joint eigenvalues (\alpha_i, \beta_i) are the pairs where v_i is a simultaneous eigenvector, and the pencil's finite eigenvalues are precisely \alpha_i / \beta_i for \beta_i \neq 0, with infinite eigenvalues corresponding to \beta_i = 0. This relation underscores how commutativity aligns the individual spectra into a unified generalized spectrum.^[26] Commuting pencils form a subclass where the Kronecker canonical form simplifies significantly, particularly with no singular blocks (such as right or left minimal indices) if both A and B are full rank, ensuring the pencil is regular and decomposes solely into Jordan blocks. This absence of singular structure arises because commutativity preserves the invariance of common eigenspaces, avoiding the rectangular blocks typical in non-commuting singular pencils.^[26]

Quadratic and Higher-Order Pencils

A quadratic matrix pencil, also known as a quadratic eigenvalue problem (QEP), is defined as Q(\lambda) = \lambda^2 A + \lambda B + C, where A, B, C are n \times n matrices over the complex numbers, and the goal is to find eigenvalues \lambda and corresponding eigenvectors x \neq 0 satisfying Q(\lambda) x = 0.^[27] This generalizes the linear matrix pencil A + \lambda B by incorporating a higher-degree polynomial structure. For higher-order pencils, the formulation extends to a polynomial eigenvalue problem (PEP) of degree m > 2, given by P(\lambda) = \sum_{k=0}^m \lambda^k A_k, where each A_k is an n \times n matrix, and eigenvalues satisfy \det(P(\lambda)) = 0 with P(\lambda) x = 0. A key technique for analyzing these higher-degree pencils is linearization, which transforms the problem into an equivalent linear pencil of larger size while preserving the eigenvalues. For a quadratic pencil, a standard companion linearization yields a $2n \times 2n pencil, such as

L(\lambda) = \lambda \begin{pmatrix} A & 0 \\ 0 & I_n \end{pmatrix} + \begin{pmatrix} B & C \\ -I_n & 0 \end{pmatrix},

where I_n is the n \times n identity matrix; the finite eigenvalues of L(\lambda) match those of Q(\lambda), and infinite eigenvalues arise if \det(A) = 0.^[27]^[28] For higher-order PEPs, linearizations like the first Frobenius companion form extend this approach, producing an mn \times mn linear pencil that captures both finite and infinite eigenvalues through block structures involving the coefficient matrices A_k. The eigenvalues of a quadratic pencil consist of up to $2n roots (counting algebraic multiplicity) of the scalar polynomial \det(Q(\lambda)) = 0, potentially including eigenvalues at infinity if the leading coefficient matrix A is singular; for real coefficients, nonreal eigenvalues appear in complex conjugate pairs.^[27] Higher-order pencils of degree m have up to mn eigenvalues, defined via the roots of \det(P(\lambda)) and including infinite eigenvalues based on the ranks of the leading coefficients. Canonical forms for higher-order pencils are more intricate than for linear cases. For polynomial matrices, the Smith normal form under unimodular equivalence provides a diagonal structure with invariant polynomials as diagonal entries, revealing the elementary divisors and partial multiplicities of eigenvalues. For regular quadratic pencils (where \det(Q(\lambda)) is not identically zero), linearization allows reduction to the Jordan canonical form of the associated linear pencil, yielding Jordan-like blocks for finite and infinite eigenvalues.^[27] Generalized Kronecker canonical forms exist for singular higher-order cases but involve complex block decompositions beyond the linear pencil's structure. Higher-degree pencils present significant challenges, as the computational complexity grows with the degree m due to the enlarged size of linearizations (from n to mn), and ill-conditioning can arise from perturbations in the coefficient matrices, amplifying errors in eigenvalue computations.^[27] As an illustrative example, consider the quadratic pencil Q(\lambda) = \lambda^2 I_n - \lambda A + B, where A and B are n \times n matrices; its linearization via the companion form

L(\lambda) = \lambda \begin{pmatrix} I_n & 0 \\ 0 & I_n \end{pmatrix} + \begin{pmatrix} -A & B \\ -I_n & 0 \end{pmatrix}

shares the same eigenvalues as Q(\lambda), which can then be found by solving the generalized eigenvalue problem for L(\lambda).^[27]^[28]

Applications

In Eigenvalue Problems

The generalized eigenvalue problem seeks scalar values \lambda and nonzero vectors v satisfying A v = \lambda B v, where A and B are square matrices of the same order; this is equivalently formulated as finding the eigenvalues of the matrix pencil A - \lambda B. When B = I, the identity matrix, the problem reduces to the standard eigenvalue problem A v = \lambda v. This generalization is particularly valuable in contexts requiring non-orthogonal bases or indefinite inner products, such as certain formulations in quantum chemistry where non-orthogonal atomic orbitals lead to overlap matrices B \neq I.^[29] In structural mechanics, vibration modes are modeled via the generalized eigenvalue problem (K - \lambda M) x = 0, where M is the mass matrix, K is the stiffness matrix, and \lambda = \omega^2 corresponds to the squared natural frequencies \omega. The eigenvectors x represent the mode shapes, enabling analysis of dynamic responses in engineering structures. For regular pencils, where \det(A - \lambda B) is not identically zero, a complete set of eigenvectors and generalized eigenvector chains spans the entire vector space, ensuring a full modal decomposition analogous to the Jordan form for standard problems. Challenges arise when B is singular, introducing infinite eigenvalues that must be handled through deflation techniques, such as shifting the pencil to A - \mu B for a finite \mu or projecting onto deflating subspaces to isolate finite eigenvalues while preserving numerical conditioning.^[30] These methods improve computational stability by avoiding ill-conditioned transformations near infinity.^[31] The eigenvalues of the pencil remain invariant under strict equivalence transformations A' = P A Q and B' = P B Q for nonsingular P and Q, linking solutions to the pencil's spectral properties without altering the characteristic polynomial.

In Control Theory

In control theory, matrix pencils are fundamental to the analysis and synthesis of descriptor systems, which model dynamic systems with algebraic constraints, such as E \dot{x} = A x + B u, where E is a singular matrix. The pencil sE - A encapsulates the system's generalized eigenvalues, known as poles, that govern the dynamic behavior and response characteristics. For stability analysis, the regularity of the pencil—ensuring \det(sE - A) is not identically zero—guarantees the system's well-posedness, while infinite eigenvalues signal the presence of impulsive modes that may lead to unbounded solutions unless controlled. In controllable descriptor systems, state feedback allows arbitrary assignment of the finite poles to achieve desired stability and performance, transforming the closed-loop pencil to place eigenvalues in specified regions of the complex plane. Output feedback extends this capability by solving for feedback matrices that modify the pencil's spectrum while respecting observability constraints.^[32]^[33] A practical example arises in electrical circuits, where descriptor models capture networks with capacitors and inductors leading to singular E; the pencil's eigenvalues then yield the natural frequencies essential for resonance and transient analysis. These applications trace back to the 1970s, when singular systems gained prominence in modern control theory for handling constrained mechanical and electrical dynamics.^[34]

In Signal Processing

In signal processing, matrix pencils are employed for high-resolution parameter estimation in noisy signals, particularly for decomposing sums of exponentially damped sinusoids into their frequencies, damping factors, and amplitudes. The matrix pencil method (MPM), introduced in the late 1980s, constructs a Hankel matrix pencil from the signal data to estimate these parameters by solving a generalized eigenvalue problem. This approach exploits the shift-invariance property of the signal's exponential components, forming two Hankel matrices Y_1 and Y_2 from the data samples, where the pencil Y_2 - \lambda Y_1 yields eigenvalues corresponding to the signal poles. The signal model underlying MPM is typically y(t) = \sum_{k=1}^K a_k \exp((\sigma_k + j 2\pi f_k) t) + n(t), where a_k is the amplitude, \sigma_k the damping factor, f_k the frequency, and n(t) noise. For discrete samples with sampling interval \Delta t, the eigenvalues z_k of the pencil satisfy z_k = \exp((\sigma_k + j 2\pi f_k) \Delta t), from which frequencies and dampings are recovered via f_k = \frac{1}{2\pi \Delta t} \Im(\ln z_k) and \sigma_k = \frac{1}{\Delta t} \Re(\ln z_k). MPM offers superior resolution and efficiency compared to the fast Fourier transform (FFT) for closely spaced frequencies, as it avoids spectral leakage and achieves super-resolution through eigenvalue separation, while being robust to noise by selecting the largest singular values to estimate signal subspace dimension. The ESPRIT algorithm extends this pencil structure to subspace-based direction-of-arrival (DOA) estimation in array processing, leveraging rotational invariance in uniform linear arrays (ULAs).^[35] By forming subarrays and constructing a pencil from the signal subspace of the array covariance matrix, ESPRIT solves for the rotation matrix whose eigenvalues relate to the DOA angles via \sin \theta_k = \frac{\lambda}{2 \pi d} \arg(\lambda_k), where d is sensor spacing, \lambda is the signal wavelength, and \lambda_k is the eigenvalue.^[35] For a ULA example, impinging plane waves produce a covariance matrix whose eigendecomposition isolates the signal subspace; the pencil on shifted subarrays then directly yields DOAs without searching a spectrum, enabling real-time processing.^[35] Since its inception in the 1980s, the matrix pencil framework has been extended to multidimensional signals, such as 2D frequency estimation in image processing via enhanced matrix pencils that handle rectangular data arrays.^[36] Further developments include adaptations for broadband sources, where frequency-dependent focusing or time-domain decomposition preprocesses signals before applying the pencil to estimate DOAs across wide bandwidths.^[37]

References

[1]
[PDF] On the Characteristic Polynomial of Regular Linear Matrix Pencil
In this paper, we present an algorithm for algebraic construction of the characteristic polynomial of a regular linear pencil. The main theorem reveals a ...
[2]
[PDF] the generalized eigenvalue problem for nonsquare pencils using a ...
This work focuses on nonsquare matrix pencils A − λB, where A, B ∈ Mm×n and m>n. Traditional methods for solving such nonsquare generalized eigenvalue problems ...
[3]
Matrix pencils: Theory, applications, and numerical methods
This paper surveys nearly all of the publications that have appeared in the last twenty years on the theory of and numerical methods for linear pencils.
[4]
The computation of Kronecker's canonical form of a singular pencil
We develop stable algorithms for the computation of the Kronecker structure of an arbitrary pencil.
[5]
[PDF] PSEUDOSPECTRA OF MATRIX PENCILS FOR TRANSIENT ...
To understand the solution of a linear, time-invariant differential-algebraic equation. (DAE), one must analyze a matrix pencil (A, E) with singular E. Even ...
[6]
[PDF] A duality relation for matrix pencils with application to linearizations
Oct 17, 2011 · linear algebra that if we complete a basis of (A) to a basis of Cn d ... matrix pencil, and let M(λ, µ) := µM0 − λM1 ∈ C. N,N. 1. [λ, µ] ...
[7]
[PDF] applications of the theory of matrices
Singular Bundles of Matrices. 2. Regular pencils of matrices. 3. Singular pencils. Theorem on reduction. . 4. Canonical form of a singular pencil of matrices.
[8]
The Theory Of Matrices Vol 2 : F. R. Gantmacher - Internet Archive
Dec 17, 2022 · The book is based on lecture courses on the theory of matrices and its applications that the author has given several times in the course of the last seventeen ...Missing: pencil | Show results with:pencil
[9]
An explicit description of the irreducible components of the set of ...
May 1, 2017 · The normal rank of a matrix pencil Q ( λ ) , denoted by nrank Q , is the rank of Q ( λ ) considered as a matrix over C ( λ ) . In other words, ...
[10]
[PDF] Basic Theory of Algebraic Eigenvalue Problems
When A − λB is regular, p(λ) = det(A − λB) is called the characteristic polynomial of A − λB. ... Weierstrass canonical form for a regular matrix pencil is a ...<|control11|><|separator|>
[11]
None
### Summary of Generalized Eigenvalue Problems (https://www.mcc.uiuc.edu/summerschool/2012/talks/05_05_Generalized%20Eigenvalue%20problems.pdf)
[12]
[PDF] Duality of matrix pencils, Wong chains and linearizations
Jul 31, 2014 · Wong chains are essentially a generalization of Jordan chains ... reconstruct left minimal indices and generalize left eigenvectors and Jordan ...
[13]
Singular Matrix Pencils and the QZ Algorithm, Update
Jan 10, 2023 · The QZ algorithm can be used to detect and analyze exceptional instances of the problem known as singular pencils.<|control11|><|separator|>
[14]
[PDF] arXiv:1311.2130v4 [math.NA] 2 Dec 2014
Dec 2, 2014 · [Weierstrass canonical form for a matrix pencil] Let A− λB be a regular pair. Then there exist nonsingular matrices P and Q such that. P AQ ...
[15]
[PDF] arXiv:1503.05035v4 [math.NA] 17 May 2015
May 17, 2015 · Below is a generalization of the Jordan canonical form to the regular ... where Jd is a d × d matrix in Jordan canonical form with its diagonal ...
[16]
[PDF] Derivation of the Maximum a Posteriori Estimate for Discrete Time ...
Mar 16, 2014 · The Kronecker canonical form transformation (KCF) for singular matrix pencils λE−A was developed by the German mathematician Leopold Kronecker ...
[17]
Kronecker Canonical Form - NetLib.org
A Kronecker canonical form (KCF) which describes the generalized eigenvalues and generalized eigenspaces of a pencil $A - \lambda B$ in full detail.
[18]
[PDF] The Computation of Kronecker's Canonical Form of a Singular Pencil
We develop stable algorithms for the computation of the Kronecker structure of an arbitrary pencil. This problem can be viewed as a generalization of the ...
[19]
On the Kronecker Canonical Form of Singular Mixed Matrix Pencils
This paper investigates the Kronecker canonical form of matrix pencils under the genericity assumption that the set of nonzero entries is algebraically ...
[20]
[PDF] Two classical theorems on commuting matrices
Simple proofs are given of the following classical theorems: (1) An arbitrary set of commuting matrices may be simultaneously brought to triangular form by a ...
[21]
[PDF] the theory of - matrices - School of Mathematics
The present book is an attempt to fill this gap in the mathematical literature. The book is based on lecture courses on the theory of matrices and its.
[22]
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### Summary of Properties of Pencils from Commuting Diagonalizable Matrices
[23]
[PDF] arXiv:2402.00673v2 [math.SP] 16 Mar 2024
Mar 16, 2024 · Before we will focus on pencils and matrix polynomials with commuting coefficients let us extend a result posted in [11] (Theorem 5.4). Theorem ...
[24]
The Quadratic Eigenvalue Problem | SIAM Review
We survey the quadratic eigenvalue problem, treating its many applications, its mathematical properties, and a variety of numerical solution techniques.
[25]
The generalized eigenvalue problem in quantum chemistry
The matrix form of the molecular orbital equations is that of a generalized eigenvalue equation, when the basis functions are non-orthogonal.Missing: mechanics | Show results with:mechanics
[26]
A deflation method for regular matrix pencils - ScienceDirect.com
Nov 15, 2011 · If we consider a regular matrix pencil containing finite and infinite forms, then we can deflate it with 3 possibilities, where an eigenpencil ...Missing: conditioning | Show results with:conditioning
[27]
[2305.13118] Analysis of eigenvalue condition numbers for a ... - arXiv
May 22, 2023 · The numerical solution of the generalized eigenvalue problem for a singular matrix pencil is challenging due to the discontinuity of its ...
[28]
Feedback and pole placement in descriptor variable Systems
Abstract. The effects and uses of applying linear feedback to continuous time descriptor systems are studied. Structural changes resulting from feeding back ...
[29]
An intermediate algorithm for pole placement by output feedback in ...
May 1, 1992 · An intermediate algorithm for pole placement by output feedback in descriptor systems ... Cobb, 1981,Int. J. Control,33, 1135.Google ...
[30]
A survey of linear singular systems
Sep 12, 1985 · This paper is a brief historical review of linear singular systems, followed by a survey of results on their solution and properties.
[31]
ESPRIT-estimation of signal parameters via rotational invariance ...
Abstract: An approach to the general problem of signal parameter estimation is described. The algorithm differs from its predecessor in that a total ...Missing: URL | Show results with:URL
[32]
https://www.tandfonline.com/doi/abs/10.1080/00207178108922981
[33]
[PDF] Application of the Matrix Pencil Approach to Direction Finding - DTIC
ABSTRACTNum* 2m. The Matrix Pencil Approach (1) was shown to be an effective and efficient method for estimating the angles of arrival of multiple ...