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Perturbation of monic matrix polynomials

This paper demonstrates that for monic matrix polynomials with coefficients depending continuously and semialgebraically on a parameter, key spectral properties like the spectrum, pseudospectrum, and numerical ranges are Hölder continuous, while eigenvalues and Jordan pairs vary analytically on a finite decomposition of the parameter space into semialgebraic submanifolds.

Original authors: Cong Trinh Le, Gue Myung Lee, Yongdo Lim, Tien Son Pham

Published 2026-03-03

📖 5 min read🧠 Deep dive

CC BY 4.0

Original authors: Cong Trinh Le, Gue Myung Lee, Yongdo Lim, Tien Son Pham

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 you are an architect designing a skyscraper. The stability of your building depends on the materials you use (steel, glass, concrete) and how they are arranged. In the world of mathematics, matrix polynomials are like these skyscrapers. They are complex structures built from numbers (matrices) and a variable (let's call it $\lambda$ ).

This paper is about what happens when you slightly tweak the materials of your skyscraper. What if the steel gets a tiny bit softer, or the concrete a tiny bit denser? Does the building collapse? Does it wobble uncontrollably? Or does it stay mostly the same, just shifting a little bit?

Here is the breakdown of the paper's findings, translated into everyday language:

1. The Setup: The "Skyscraper" and the "Dial"

The authors are studying a specific type of mathematical building called a monic matrix polynomial.

The Building: It's a formula that looks like $A_d \lambda^d + \dots + A_0$ .
The "Monic" Rule: The top part of the building (the coefficient of the highest power) is fixed as the "Identity Matrix." Think of this as the foundation being made of unbreakable, perfect steel. This simplifies the math so they don't have to worry about the foundation crumbling.
The Dial ( $u$ ): The other parts of the building (the lower coefficients) aren't fixed. They depend on a control knob or a "dial" called $u$ . You can turn this dial to change the materials slightly.

2. The Big Question: How Stable is the Building?

The authors ask: If I turn the dial $u$ just a tiny bit, how much does the "behavior" of the building change?

They look at four specific ways to measure the building's behavior:

The Spectrum (The "Resonant Frequencies"): These are the specific points where the building might vibrate or resonate. In math, these are the "roots" or solutions to the equation.
The Pseudospectrum (The "Safety Zone"): This is a safety buffer. It asks: "If I make a tiny mistake in my measurements or if the materials shift slightly, where could the resonant frequencies end up?" It's a fuzzy cloud around the exact frequencies.
The Numerical Range (The "Energy Field"): This measures the range of energy the building can hold. It's like a map of all the possible forces the structure can withstand.
The Joint Numerical Range (The "Multi-Dimensional Map"): This is a more complex map that tracks how all the different parts of the building interact with each other simultaneously.

3. The Main Discovery: "Hölder Continuity" (The Rubber Band Effect)

The paper proves that these behaviors are stable.

Imagine the relationship between your dial ( $u$ ) and the building's behavior (like the resonant frequencies) is connected by a rubber band.

If you pull the dial a tiny bit, the rubber band stretches a tiny bit.
The behavior doesn't snap or jump wildly. It moves smoothly.
The authors call this Hölder continuity. In plain English, it means: "If you change the input by a small amount, the output changes by a predictable, small amount."

They even found that for most of the time, if you turn the dial smoothly, the building's behavior changes smoothly too.

4. The "Smooth Road" vs. The "Bumpy Road"

The authors discovered something fascinating about the "dial" (the parameter space).

The Bumpy Road: If you look at the entire range of possible settings for your dial, there are some weird, jagged spots where the rules change. Maybe the number of resonant frequencies suddenly changes, or the shape of the safety zone gets weird.
The Smooth Road: However, they proved that you can break the entire dial into finite pieces. On almost every piece (specifically, on "analytic semialgebraic submanifolds"), the building behaves perfectly smoothly.
- Analogy: Imagine driving a car. Most of the road is a smooth highway where you can cruise at a steady speed. There are a few specific intersections or construction zones (the "submanifolds") where you have to slow down or change gears. But once you pass those zones, you are back on a smooth, predictable road.

5. Why Does This Matter?

In the real world, nothing is perfect.

Engineering: When engineers design bridges or airplanes, the materials aren't perfect, and the wind isn't constant. They need to know: "If the wind speed varies by 1%, will the bridge's vibration frequency jump by 50% (bad) or 0.1% (good)?"
Control Systems: If you are programming a robot, you need to know that if your sensors are slightly off, the robot won't suddenly spin out of control.

This paper gives mathematicians and engineers a guarantee. It says: "As long as your system follows these rules (monic matrix polynomials), you can trust that small changes in your inputs will only cause small, predictable changes in your outputs. The system is robust."

Summary in a Nutshell

The authors took a complex mathematical structure (a matrix polynomial) and showed that it is tough. Even if you tweak the ingredients slightly, the structure doesn't fall apart or behave erratically. It shifts gently and predictably. They mapped out exactly where the "smooth roads" are and proved that the building's behavior is mathematically stable, giving us confidence to use these structures in real-world applications like control systems and physics.

1. Problem Statement

The paper investigates the stability of spectral properties of matrix polynomials under structured perturbations. Unlike classical perturbation theory, which often considers arbitrary perturbations of bounded norm, this work focuses on perturbations induced by a parameter vector $u \in \mathbb{C}^p$ .

The object of study is a family of matrix polynomials:
$P_u(\lambda) = A_d(u)\lambda^d + A_{d-1}(u)\lambda^{d-1} + \dots + A_0(u)$
where the coefficients $A_k(u)$ depend continuously and semialgebraically on the parameter $u$ . The authors assume the polynomial is monic (i.e., $A_d(u) \equiv I$ , the identity matrix), though they note many results extend to locally nonsingular leading coefficients.

The core question is: How stable are the following set-valued maps with respect to the parameter $u$ ?

Spectrum ( $\sigma(u)$ ): The set of roots of $\det P_u(\lambda) = 0$ .
$\epsilon$ -Pseudospectrum ( $\Lambda_\epsilon(u)$ ): The set of $\lambda$ where $\det P_v(\lambda) = 0$ for some parameter $v$ within distance $\epsilon$ of $u$ .
Numerical Range ( $W(u)$ ): The set $\{\lambda \in \mathbb{C} \mid x^* P_u(\lambda) x = 0, \|x\|=1\}$ .
Joint Numerical Range ($JW(u)$): The image of the unit sphere under the map $x \mapsto (x^*A_0(u)x, \dots, x^*A_d(u)x)$ .
Spectral Radius ( $r(u)$ ): The maximum modulus of the eigenvalues.

2. Methodology

The authors employ tools from semialgebraic geometry to analyze the continuity and regularity of these spectral objects.

Semialgebraic Framework: The coefficients $A_k(u)$ are assumed to be continuous semialgebraic functions. Consequently, the sets defined by the spectral properties (spectrum, pseudospectrum, numerical ranges) are shown to be semialgebraic sets via the Tarski–Seidenberg theorem (which ensures that projections of semialgebraic sets remain semialgebraic).
Lojasiewicz Inequality: A central tool used is the Lojasiewicz inequality for continuous semialgebraic maps. This inequality guarantees that on compact sets, such maps satisfy a Hölder continuity condition: $\|f(x) - f(y)\| \leq c \|x - y\|^\alpha$ .
Cell Decomposition: The authors utilize the Cell Decomposition Theorem to partition the parameter space $\mathbb{C}^p$ into a finite union of analytic semialgebraic submanifolds. On each submanifold, the spectral objects behave analytically.
Linearization: For the analysis of Jordan pairs, the matrix polynomial is linearized into a companion matrix form, reducing the problem to the stability of eigenvalues and Jordan structures of a linear matrix pencil.

3. Key Contributions and Results

A. Hölder Continuity of Set-Valued Maps

The primary result is that the set-valued maps for the spectrum, pseudospectrum, and numerical ranges are Hölder continuous with respect to the parameter $u$ .

Spectrum ( $\sigma$ ): The map $u \mapsto \sigma(u)$ is Hölder continuous. The proof relies on bounding the distance between roots of perturbed monic polynomials using the determinant and matrix norms.
$\epsilon$ -Pseudospectrum ( $\Lambda_\epsilon$ ): The map is Hölder continuous with respect to both the parameter $u$ and the radius $\epsilon$ . The boundary $\partial \Lambda_\epsilon(u)$ is shown to lie on an algebraic curve, implying it has only a finite number of singularities (cusps or self-intersections).
Numerical Range ( $W$ ) and Boundary ( $\partial W$ ): The map $u \mapsto W(u)$ and its boundary are Hölder continuous. The boundary is also shown to be a semialgebraic set lying on an algebraic curve with finite singularities.
Joint Numerical Range ($JW$): The map $u \mapsto JW(u)$ is Hölder continuous.

B. Explicit Exponents and Generic Regularity

The paper provides explicit Hölder exponents for these maps on a generic (open and dense) semialgebraic subset $U \subset \mathbb{C}^p$ :

Spectrum ( $\sigma$ ): Hölder continuous with degree $1/(nd)$.
$\epsilon$ -Pseudospectrum ( $\Lambda_\epsilon$ ): Hölder continuous with degree $1/(nd)$.
Numerical Range ( $W$ ) and its boundary: Hölder continuous with degree $1/d$ .
Joint Numerical Range ($JW$): Lipschitz continuous (degree 1) on the generic set.
Spectral Radius ( $r(u)$ ): Analytic on the generic set.

C. Analyticity of Jordan Pairs

The paper addresses the stability of the Jordan structure (eigenvalues and Jordan chains).

Theorem 4.3: The parameter space $\mathbb{C}^p$ can be decomposed into a finite union of analytic semialgebraic submanifolds.
Result: On each submanifold:
1. The number of distinct eigenvalues and the sizes of their corresponding Jordan blocks are constant.
2. The eigenvalues $\lambda_i(u)$ are analytic functions of $u$ .
3. The matrices forming the Jordan pairs (the eigenvector matrices and Jordan blocks) depend analytically on $u$ .

4. Significance and Implications

Structured Perturbation Theory: The work bridges the gap between classical unstructured perturbation theory and real-world applications where coefficients depend on physical or design parameters. It formalizes the stability of spectral objects under these specific, structured variations.
Geometric Regularity: By proving that the boundaries of pseudospectra and numerical ranges lie on algebraic curves with finite singularities, the paper provides strong geometric guarantees useful for numerical algorithms and visualization.
Generic Smoothness: The decomposition of the parameter space into analytic submanifolds clarifies that while global analyticity of eigenvalues is impossible (due to branching at collisions), local analyticity holds almost everywhere (on an open dense set). This is crucial for optimization and control problems involving matrix polynomials.
Extension to Non-Monic Cases: The authors demonstrate that the monic assumption is primarily for exposition; the results largely hold if the leading coefficient $A_d(u)$ is merely locally nonsingular, broadening the applicability to general matrix polynomials.

5. Conclusion

This paper establishes a rigorous framework for understanding the stability of matrix polynomials under parameter-dependent perturbations. By leveraging semialgebraic geometry, the authors prove that spectral sets are Hölder continuous and that the underlying spectral structure (eigenvalues and Jordan pairs) varies analytically on a generic decomposition of the parameter space. These results provide a theoretical foundation for the robustness analysis of systems modeled by matrix polynomials in engineering and applied mathematics.