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On equivalence classes of dissipative Hamiltonian pencils

This paper characterizes the maximal elements and orbit closures of strict equivalence classes for dissipative Hamiltonian matrix pencils within Pokrzywa's ordering framework, while also analyzing degenerations and deriving partial results on the Kronecker canonical form for both regular and singular cases.

Original authors: Maria Dronka

Published 2026-07-31

📖 4 min read🧠 Deep dive

CC BY 4.0

Original authors: Maria Dronka

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

Imagine the universe of mathematics as a giant, bustling city where every object is a "matrix pencil." Think of a pencil not as a writing tool, but as a dynamic equation that changes shape depending on a variable, like a chameleon shifting its colors. In this city, some pencils are "regular," meaning they are well-behaved and predictable, while others are "singular," acting like wildcards that can do almost anything. Mathematicians love to group these pencils into "orbits." If you can turn one pencil into another just by stretching, shrinking, or rotating it (without tearing it apart), they live in the same orbit. It's like saying two different-looking Lego structures are actually the same if they are built from the exact same set of blocks, just rearranged.

Now, zoom in on a very special neighborhood in this city called "dissipative Hamiltonian systems." These are the energy managers of the mathematical world. They describe how real-world machines, like swinging pendulums or electrical circuits, lose energy to friction or resistance over time. These systems have strict rules: they must always lose energy (or stay the same), never magically create it. The big question mathematicians have been asking is: "If we nudge these energy-managing pencils just a tiny bit, do they stay in their special neighborhood, or do they tumble out into the chaotic wild?" Understanding this helps engineers design safer bridges, more stable power grids, and better robots, because it tells us how much a system can be disturbed before it breaks its fundamental laws.

In this paper, Maria Dronka takes a deep dive into the "neighborhood rules" for these special energy pencils. She uses a clever map called the "Kronecker canonical form," which is like a DNA test for pencils, breaking them down into their simplest, unchangeable building blocks. Her main discovery is a set of strict boundaries on how these pencils can change. She proves that if you have a pencil where the energy rules are set to a specific, simple standard (where a certain matrix is just the identity, or "1"), it cannot accidentally morph into a "wild" pencil unless its overall size (rank) changes. It's like saying a perfectly balanced seesaw can't suddenly turn into a spinning top unless you add or remove weight entirely.

However, the paper also reveals a fascinating twist. If you relax the rules and allow the energy system to be "singular" (meaning it has some broken or missing parts), the story changes. Dronka shows that in this messy, singular state, the pencils can suddenly sprout all kinds of strange new building blocks that were previously forbidden. They can grow into almost any shape, but not any shape—they still have to follow a hidden code. For instance, she proves that even in this chaotic state, these pencils must always have at least one "zero" in their right-hand side structure, a specific constraint that keeps them from becoming completely random.

The author also identifies the "king" pencils of this neighborhood—the most generic, robust forms that sit at the top of the hierarchy. These are the ultimate versions that can't be improved upon without breaking the rules. She maps out exactly what these kings look like, showing they are made of specific combinations of simple blocks. Furthermore, she draws a detailed map of the "orbit closures," which are like the neighborhoods surrounding these kings. She shows that sometimes, the neighborhood of a perfect energy pencil is safe and contains only other perfect pencils, but other times, the neighborhood is a mix, containing pencils that have lost their perfect energy balance.

Ultimately, this paper doesn't just list rules; it draws the borders of the map. It tells us exactly which transformations are possible and which are impossible for these energy systems. While it solves the puzzle for the "perfect" (regular) case and gives a partial solution for the "broken" (singular) case, it leaves the door open for future explorers to fully decode the wild, singular territories. The work confirms that while these systems can be flexible, they are never truly free; they are always tethered by the invisible laws of energy dissipation.

Technical Summary: On Equivalence Classes of Dissipative Hamiltonian Pencils

Problem Statement
The paper investigates the strict equivalence orbit structure of matrix pencils of the form $M = \lambda E - (J - R)Q$ , where $E, J, R, Q \in \mathbb{C}^{n \times n}$ satisfy specific structural constraints: $Q^*E = E^*Q \geq 0$ , $J^* = -J$ , and $R^* = R \geq 0$ . These pencils, termed dissipative Hamiltonian (dH) pencils, arise in the modeling of linear time-invariant dissipative Hamiltonian descriptor systems.

A central challenge addressed is the behavior of these pencils under perturbation. While previous work (e.g., [18]) characterized the spectral properties of dH pencils under the assumption that the pencil $\lambda E - Q$ is regular, this paper highlights that the dH structure is not preserved under arbitrary small perturbations. Specifically, a regular dH pencil may lie in the orbit closure of a pencil of the same form where $\lambda E - Q$ is singular. Consequently, a comprehensive algebraic theory requires understanding the orbit closure hierarchy and the Kronecker canonical form (KCF) of these pencils without assuming the regularity of $\lambda E - Q$ .

Methodology
The analysis is conducted within the framework of Pokrzywa's ordering of strict equivalence orbits. The primary tool is the Kronecker canonical form (KCF), which classifies pencils up to strict equivalence. The paper utilizes a set of six elementary operations on KCF blocks (involving singular blocks $L_k, L_k^T$ , Jordan blocks $J_k^\mu$ , and nilpotent blocks $J_k^\infty$ ) that characterize the inclusion relations between orbit closures ( $O(P) \subset O(M)$ ).

The study distinguishes between two primary classes:

$\mathcal{dH}$ : Pencils strictly equivalent to a dH pencil with regular $\lambda E - Q$ .
$\mathcal{dHI}$ : Pencils strictly equivalent to a dH pencil with regular $\lambda E - Q$ and the additional constraint $Q = I$ .

The methodology involves:

Applying Pokrzywa's operations to determine maximal elements and degenerations within these classes.
Analyzing orbit stratifications, particularly in low dimensions ( $2 \times 2$ and $2 \times 3$ ), to visualize inclusion relations.
Constructing explicit matrix examples to demonstrate the realizability of specific KCF blocks within the dH structure, even when $\lambda E - Q$ is singular.

Key Contributions and Results

Restrictions on Degenerations (Case $Q=I$ ):
The paper establishes that pencils in class $\mathcal{dHI}$ cannot degenerate into pencils outside this class without a change in rank. Specifically, if $M \in \mathcal{dHI}$ and $P \in O(M)$ with $\text{nrank}(P) = \text{nrank}(M)$ , then $P \in \mathcal{dHI}$ . This result does not hold for the broader class $\mathcal{dH}$ , where rank-preserving degenerations can lead outside the class.
Characterization of Maximal Elements:
The authors characterize the maximal elements of the partial order induced by orbit closure inclusion for both $\mathcal{dH}$ and $\mathcal{dHI}$ .
- For $\mathcal{dH}$ , maximal elements consist of Jordan blocks for eigenvalues in the open left half-plane, semisimple Jordan blocks for nonzero imaginary eigenvalues, and specific configurations of zero and infinite eigenvalues (with constraints on partial multiplicities and minimal indices).
- For $\mathcal{dHI}$ , the maximal elements are similar but strictly forbid Jordan blocks of size greater than 1 for the eigenvalue zero and require zero right minimal indices.
Orbit Closure Properties:
- The paper provides conditions under which the orbit closure of a dH pencil contains only pencils strictly equivalent to dH pencils. This occurs if and only if the KCF consists solely of size-0 singular blocks and size-1 Jordan blocks with eigenvalues in the closed left half-plane (or infinity).
- A version of the classical result regarding orbit closures is derived for these classes: the closure of an orbit intersected with the class $\mathcal{dH}$ (or $\mathcal{dHI}$ ) is the union of the orbit itself and the orbits of strictly smaller dimension within that class.
KCF of Pencils $\lambda E - Q$ with $E^*Q = Q^*E \geq 0$ :
Extending results from [18], the paper fully characterizes the KCF for pencils of the form $\lambda E - Q$ (without the $J-R$ term) satisfying the Hermitian condition. Such pencils must have eigenvalues in $\mathbb{R}_{\geq 0} \cup \{\infty\}$ , all eigenvalues must be semisimple, and right minimal indices must be zero (if they exist).
Singular $\lambda E - Q$ in dH Pencils:
The paper addresses the case where $\lambda E - Q$ is singular. It proves that while such pencils can contain canonical blocks of all types and sizes, they cannot be arbitrary.
- Theorem 5.2: For sufficiently large $n$ , any specific right singular, left singular, Jordan, or nilpotent block can be realized in the KCF of a dH pencil with singular $\lambda E - Q$ .
- Observation 5.7: A crucial restriction is identified: if $\lambda E - Q$ is singular, the resulting dH pencil $M$ must possess at least one right minimal index equal to zero. This prevents certain KCF configurations (e.g., a pencil with only a right minimal index of 2) from being dH pencils with singular $\lambda E - Q$ .

Significance and Scope
The paper constitutes a foundational step toward a comprehensive algebraic theory of dH pencils, particularly in scenarios where regularity assumptions fail. By characterizing the orbit closure hierarchy and identifying the specific constraints on KCF blocks, the work clarifies the structural limitations of dissipative Hamiltonian systems under perturbation.

The authors explicitly state that a complete characterization of the structure-preserving stratification of dH pencils remains an open problem. Furthermore, while partial results on the KCF for singular cases are provided, a full characterization of the KCF for pencils of the form (1) with singular $\lambda E - Q$ is left for future research. The paper does not propose new applications or experimental setups but focuses strictly on the theoretical classification and structural properties of these matrix pencils.

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