Chapter 41A: Regular Matrix Pencils¶

Prerequisites: Eigenvalues (Ch06) · $\lambda$-matrices (Ch13B) · Generalized Eigenvalue Problems

Chapter Outline: From Single Matrices to Matrix Pairs → Definition of Matrix Pencils $A - \lambda B$ → Regular vs. Singular Pencils → Generalized Eigenvalues and Eigenvectors → The Generalized Characteristic Equation $\det(A - \lambda B) = 0$ → Weierstrass Canonical Form (Regular Case) → Handling Infinite Eigenvalues → Applications: Structural Dynamics and State-Space Analysis of Generalized Linear Systems

Extension: Matrix pencil theory is the mathematical foundation for studying generalized linear dynamical systems (such as Differential-Algebraic Equations, DAEs); it generalizes spectral theory from a single operator $A$ to the relative evolution of two operators $A$ and $B$, which is key to stability analysis in complex systems.

In classical eigenvalue problems, we study $A\mathbf{x} = \lambda \mathbf{x}$. However, in many engineering problems (such as finite element analysis), equations take the form $A\mathbf{x} = \lambda B\mathbf{x}$. Matrix Pencils $A - \lambda B$ are the tools for describing such relative spectral relationships. When the characteristic equation is not identically zero, the pencil is termed Regular. This chapter introduces the decomposition theory of regular matrix pencils and their dominant role in dynamical systems.

41A.1 Basic Concepts of Matrix Pencils¶

Definition 41A.1 (Matrix Pencil)

Given two $m \times n$ matrices $A$ and $B$, the set $\{ A - \lambda B : \lambda \in \mathbb{C} \}$ is called a Matrix Pencil.

Definition 41A.2 (Regular Pencil)

If $A$ and $B$ are square matrices of the same order and the characteristic polynomial $p(\lambda) = \det(A - \lambda B)$ is not identically zero, the pencil is Regular. Otherwise, it is Singular.

41A.2 Generalized Eigenvalues and Canonical Forms¶

Definition 41A.3 (Generalized Eigenvalues)

Scalars $\lambda$ satisfying $\det(A - \lambda B) = 0$ are finite generalized eigenvalues. If $\det(B) = 0$, the pencil may also possess infinite eigenvalues $\lambda = \infty$.

Theorem 41A.1 (Weierstrass Canonical Form)

For a regular pencil $A - \lambda B$, there exist non-singular matrices $P$ and $Q$ such that: $$P(A - \lambda B)Q = \operatorname{diag}(J - \lambda I, I - \lambda N)$$ - $J$ is in Jordan form, corresponding to finite eigenvalues. - $N$ is a nilpotent Jordan matrix, corresponding to infinite eigenvalues.

41A.3 Application in Dynamics¶

Application: Vibration Analysis

In mechanical vibrations, the governing equation is $M \ddot{x} + K x = 0$. Assuming a solution $x = e^{i\omega t} v$ leads to the generalized eigenvalue problem $Kv = \omega^2 Mv$. Here $K$ is the stiffness matrix and $M$ is the mass matrix.

Exercises¶

1. [Calculation] Find the characteristic polynomial of the pencil $A - \lambda B$ where $A = \begin{pmatrix} 1 & 0 \\ 0 & 2 \end{pmatrix}, B = \begin{pmatrix} 1 & 0 \\ 0 & 0 \end{pmatrix}$.

Solution

Steps: 1. Write the difference matrix: $A - \lambda B = \begin{pmatrix} 1-\lambda & 0 \\ 0 & 2 \end{pmatrix}$. 2. Compute the determinant: $\det(A - \lambda B) = (1-\lambda) \cdot 2 = 2 - 2\lambda$. Conclusion: The characteristic polynomial is $2 - 2\lambda$.

2. [Eigenvalues] Find the generalized eigenvalues (including $\infty$) for the previous problem.

Solution

Analysis: 1. Finite eigenvalues: Set $2 - 2\lambda = 0 \implies \lambda = 1$. 2. Infinite eigenvalues: Since $\det(B) = 0$, an infinite eigenvalue exists. 3. Degree check: Since $n=2$ but the polynomial degree is 1, the missing root is $\infty$. Conclusion: The generalized eigenvalues are $\{1, \infty\}$.

3. [Regularity] Determine if $A = \begin{pmatrix} 1 & 0 \\ 0 & 0 \end{pmatrix}, B = \begin{pmatrix} 0 & 1 \\ 0 & 0 \end{pmatrix}$ is a regular pencil.

Solution

Calculation: $A - \lambda B = \begin{pmatrix} 1 & -\lambda \\ 0 & 0 \end{pmatrix}$. $\det(A - \lambda B) = 1 \cdot 0 - (-\lambda) \cdot 0 = 0$. Conclusion: Since the determinant is identically zero, the pencil is Singular.

4. [Relation] If $B$ is invertible, how are generalized eigenvalues related to standard ones?

Solution

Conclusion: If $B$ is invertible, $A\mathbf{x} = \lambda B\mathbf{x}$ is equivalent to $(B^{-1}A)\mathbf{x} = \lambda \mathbf{x}$. In this case, all eigenvalues are finite and correspond to the standard eigenvalues of $B^{-1}A$.

5. [Infinite] What is the physical meaning of an infinite eigenvalue in a matrix pencil?

Solution

Explanation: In Differential-Algebraic Equations (DAEs), infinite eigenvalues typically correspond to algebraic constraints. They represent variables with infinitely fast response times (instantaneous response) or indicate a degeneracy in the system's dynamic order. They are described by the nilpotent part $N$ in the Weierstrass form.

6. [Calculation] Find the generalized eigenvalues of $A = \begin{pmatrix} 0 & 1 \\ 1 & 0 \end{pmatrix}$ relative to $B = I$.

Solution

Calculation: Since $B=I$, this reduces to the standard eigenvalue problem for $A$. $\det(A - \lambda I) = \lambda^2 - 1 = 0 \implies \lambda = \pm 1$.

7. [Weierstrass] In the block $I - \lambda N$ of the Weierstrass form, what properties does $N$ satisfy?

Solution

Property: $N$ is a nilpotent matrix ($N^k = O$ for some $k$). Its non-zero entries are on the super-diagonal, representing the Jordan chains associated with the infinite eigenvalue.

8. [Stability] If the real parts of all finite generalized eigenvalues are negative and there are no infinite eigenvalues, is the system stable?

Solution

Yes. This ensures that the differential part of the evolution decays and there are no impulsive terms or order mismatches caused by algebraic constraints.

9. [Diagonalization] Under what condition can two matrices $A$ and $B$ be simultaneously diagonalized?

Solution

Conclusion: If $A$ and $B$ are both Hermitian and one of them (usually $B$) is positive definite, they can be simultaneously diagonalized. This is the basis for "modal decomposition" in mechanical vibrations.

10. [Application] How are generalized eigenvalues used to find transmission zeros in control theory?

Solution

Connection: Transmission zeros are defined as the complex values $s$ that make the system matrix pencil (the Rosenbrock matrix) lose rank. This is essentially searching for generalized eigenvalues within a specific matrix pencil structure.

Chapter Summary¶

Regular matrix pencil theory is the ultimate map for generalized linear systems:

Relativity of Spectra: It elevates eigenvalues from an inherent property of one operator to a measure of interference between two, establishing a universal framework for relative evolution in physical systems.
Parsing Infinity: By introducing infinite eigenvalues and nilpotent structures, pencil theory perfectly captures abrupt changes, constraints, and singularities in continuous systems.
Standardization of Structure: The Weierstrass Canonical Form provides the ultimate reference for coordinate transformations in complex DAEs, enabling complete decoupling of dynamical modes.