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Eigenvalue stability of Hermitian and normal matrices

This paper establishes the continuity and boundedness properties of the ordered eigenvalue map acting on Sobolev spaces for Hermitian and normal matrices, demonstrating that while the map is continuous for finite $q$ , it fails to be uniformly continuous or continuous for $q=\infty$ , with extensions to applications involving singular values, condition numbers, and compact operators.

Original authors: Adam Parusiński, Armin Rainer

Published 2026-03-25

📖 5 min read🧠 Deep dive

CC BY 4.0

Original authors: Adam Parusiński, Armin Rainer

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 have a giant, magical machine that takes in a complex, shifting object (like a kaleidoscope of colors or a vibrating metal sheet) and spits out a list of its most important "vibes" or eigenvalues. In the world of mathematics, these objects are called matrices, and the "vibes" are the numbers that describe how the matrix stretches or rotates space.

This paper, written by Adam Parusiński and Armin Rainer, is essentially a study of stability. It asks a very practical question: If I make a tiny, smooth change to the input machine, how much do the output "vibes" wiggle?

Here is the breakdown of their findings using everyday analogies.

1. The Two Types of Machines

The authors look at two specific types of machines:

Hermitian Matrices: Think of these as machines that produce real numbers (like temperatures or heights). They are very orderly. You can line up their outputs from smallest to largest: 1, 2, 3, 4, 5.
Normal Matrices: These are more chaotic. They produce complex numbers (numbers with a real part and an imaginary part, like coordinates on a map). You can't easily say one is "bigger" than the other because they exist in a 2D plane. It's like trying to order a list of cities by "size" when some are big but short, and others are small but tall.

2. The Smoothness Test (The "Rubber Sheet" Analogy)

The researchers are testing the machine using a specific type of smoothness called Sobolev spaces ( $W^{1,q}$ ).

The Analogy: Imagine the input machine is a rubber sheet.
- $L^\infty$ (Lipschitz): This measures how much the sheet stretches right now. If you pull it, how far does it move?
- $W^{1,q}$ : This measures how much the sheet stretches and how fast the stretching changes as you move your hand across it. It's about the "smoothness of the motion."

The Big Discovery:
The authors found that if you move your hand smoothly across the rubber sheet (changing the matrix smoothly), the list of "vibes" (eigenvalues) also changes smoothly in a specific way.

For Hermitian (Real) Matrices: The list of numbers changes continuously. If you nudge the machine slightly, the list of numbers shifts slightly.
For Normal (Complex) Matrices: Since you can't order the complex numbers perfectly, the "list" is actually a cloud of points. The paper proves that this entire cloud moves smoothly as a group, even if individual points inside the cloud swap places or spin around. They treat this cloud as a "multi-valued function" (like a cloud of bees that moves together, even if individual bees buzz around).

3. The "Gotcha": Smoothness vs. Uniformity

Here is the twist in the story. The authors found that while the machine is stable, it isn't uniformly stable.

The Metaphor: Imagine a car suspension system.
- Stable: If you hit a small bump, the car doesn't fly off the road. The ride is smooth.
- Not Uniformly Stable: However, if you hit a bump at a very specific, tricky angle (like a sharp corner in the road), the suspension might react violently, even if the bump itself was small.

In math terms:

If you have a sequence of machines getting closer and closer to a target, the eigenvalues will get closer and closer.
BUT, you cannot guarantee that every possible small change results in a proportionally small change in the eigenvalues. There are "tricky spots" where the eigenvalues are very sensitive to tiny, high-frequency wiggles in the input.

4. Why This Matters (Real World Applications)

Why should a regular person care about the stability of matrix numbers? Because these matrices model almost everything in physics and engineering.

Singular Values (The "Strength" of a Signal): In image processing or data compression (like JPEGs), we look at the "strength" of different parts of an image. This paper proves that if your image data changes smoothly, the "strength" of the image features changes smoothly too. This is crucial for making sure algorithms don't crash when data gets noisy.
Condition Numbers (The "Fragility" of a Problem): In engineering, some problems are "ill-conditioned," meaning a tiny error in measurement leads to a huge error in the result (like a house of cards). This paper helps us understand how the "fragility" of a system changes as the system itself changes.
Quantum Mechanics: In quantum physics, the "eigenvalues" are the energy levels of a particle. If the environment (the matrix) changes smoothly over time, this paper guarantees that the energy levels also evolve smoothly, which is vital for understanding quantum systems.
Surface Area: They even used this to calculate the surface area of graphs formed by these eigenvalues, which has applications in geometry and material science.

5. The "Eigenvector" Problem

There is one catch the authors mention. While the numbers (eigenvalues) behave well, the directions (eigenvectors) often do not.

The Metaphor: Imagine a spinning top. The speed of the spin (eigenvalue) might be very stable. But the axis the top is spinning on (eigenvector) might suddenly flip 180 degrees if the top wobbles just right. The paper confirms that while the "speed" is smooth, the "axis" can be jerky and discontinuous.

Summary

This paper is a rigorous proof that nature's "vibes" (eigenvalues) are generally well-behaved and smooth, even when the underlying system is complex. It tells us that we can trust our mathematical models to evolve smoothly, provided we are careful about the specific type of smoothness we are measuring. It's a safety certificate for the mathematical foundations of physics, engineering, and data science.

1. Problem Statement

The paper investigates the regularity and stability of eigenvalues as functions of the underlying matrix entries, specifically within the context of Sobolev spaces ( $W^{1,q}$ ) and Lipschitz spaces ( $C^{0,1}$ ).

The Core Question: Given a family of Hermitian or normal matrices $A(x)$ depending on a parameter $x$ in a domain $U \subseteq \mathbb{R}^m$ , how regular are the eigenvalues $\lambda(x)$ ?
The Challenge: While Weyl's perturbation theorem guarantees that the map from a matrix to its ordered eigenvalues is Lipschitz continuous (with constant 1) in the operator norm, this does not automatically imply that the composition map (superposition) preserves Sobolev regularity or continuity in stronger topologies.
- In the Hermitian case, eigenvalues can be ordered ( $\lambda_1 \le \dots \le \lambda_d$ ), defining a map $\lambda^\uparrow: \text{Herm}(d) \to \mathbb{R}^d$ .
- In the Normal case, eigenvalues are complex and cannot be globally ordered; they must be treated as unordered tuples (multivalued functions) in the space of unordered $d$ -tuples, denoted $\text{Ad}(\mathbb{C})$ .
Specific Gaps: Previous work established continuity for hyperbolic polynomials (real roots) in Sobolev spaces. This paper extends these results to the matrix setting, addressing whether the "characteristic map" (mapping a matrix family to its eigenvalue family) is continuous, uniformly continuous, or Hölder continuous in various topologies.

2. Methodology

The authors employ a sophisticated blend of perturbation theory, geometric analysis, and functional analysis:

Lipschitz Perturbation Theory: They rely fundamentally on Weyl's perturbation theorem (for Hermitian matrices) and the Hoffman-Wielandt theorem (for normal matrices), which establish that the eigenvalue map is 1-Lipschitz with respect to the Frobenius (2-) norm.
Uniform Block-Diagonalization: A central technical tool is a splitting lemma (derived from Hensel's lemma and Nash geometry). This allows for the local uniform unitary block-diagonalization of Hermitian and normal matrices.
- Whenever eigenvalues do not coincide, the matrix can be locally diagonalized into smaller blocks via a unitary transformation $U(x)$ that is smooth (Nash) with respect to the matrix entries.
- This reduces the problem of analyzing $d \times d$ matrices to analyzing smaller blocks, facilitating an inductive argument on the dimension $d$ .
Multivalued Sobolev Functions (Almgren's Framework): For normal matrices, eigenvalues are treated as functions taking values in the metric space of unordered tuples $\text{Ad}(\mathbb{C})$ . The authors utilize Almgren's bi-Lipschitz embedding $\Delta: \text{Ad}(\mathbb{C}) \to \mathbb{R}^N$ to define Sobolev spaces of these multivalued functions.
Vitali's Convergence Theorem: To prove continuity in $W^{1,q}$ , the authors establish uniform integrability of the derivatives of the eigenvalues. They use de la Vallée Poussin's criterion to show that the sequence of derivatives is uniformly integrable, allowing the application of Vitali's theorem to upgrade convergence in measure to convergence in $L^q$ .
Slicing Arguments: For the multiparameter case ( $m > 1$ ), they reduce the problem to the one-parameter case ( $m=1$ ) by restricting the functions to line segments parallel to coordinate axes and using Fubini/Tonelli theorems.

3. Key Contributions and Results

A. Hermitian Matrices (Ordered Eigenvalues)

Let $E: W^{1,q}(U, \text{Herm}(d)) \to W^{1,q}(U, \mathbb{R}^d)$ be the map $A \mapsto \lambda^\uparrow \circ A$ .

Continuity in $W^{1,q}$ ( $1 \le q < \infty$ ): The map $E$ is continuous. If $A_n \to A$ in $W^{1,q}$ , then $\lambda^\uparrow(A_n) \to \lambda^\uparrow(A)$ in $W^{1,q}$ .
Non-Uniform Continuity: The map is not uniformly continuous on bounded sets for any $q \in [1, \infty)$ .
Lipschitz Case ( $q = \infty$ ):
- The map is not continuous from $C^{0,1}$ to $C^{0,1}$ (the target topology is too strong).
- However, it is continuous from $C^{0,1}$ to $W^{1,q}$ for any $q < \infty$ .
- It is also continuous from $C^{0,1}$ to the Hölder space $C^{0,\alpha}$ for any $0 < \alpha < 1$ , but not for $\alpha = 1$ .
Almost Everywhere Convergence: If $A_n \to A$ in $C^{0,1}$ , then the derivatives of the eigenvalues converge pointwise almost everywhere: $\partial_j \lambda^\uparrow(A_n)(x) \to \partial_j \lambda^\uparrow(A)(x)$ .

B. Normal Matrices (Unordered Eigenvalues)

Let $E_u: W^{1,q}(U, \text{Norm}(d)) \to W^{1,q}(U, \text{Ad}(\mathbb{C}))$ be the map $A \mapsto \Lambda \circ A$ .

Continuity in $W^{1,q}$ : The map is continuous for all $1 \le q < \infty$ .
Multivalued Interpretation: The eigenvalues are naturally interpreted as multivalued Sobolev functions (in the sense of Almgren/De Lellis-Spadaro). The convergence is defined via the metric on $\text{Ad}(\mathbb{C})$ and its embedding.
Lipschitz Case: Similar to the Hermitian case, the map is continuous from $C^{0,1}$ to $W^{1,q}$ and $C^{0,\alpha}$ ( $\alpha < 1$ ), but not to $C^{0,1}$ .

C. Applications and Extensions

The results extend to several important quantities:

Singular Values: Since singular values are eigenvalues of a related Hermitian matrix, the stability results apply directly. The map $A \mapsto \sigma(A)$ is continuous in $W^{1,q}$ .
Condition Numbers: The condition number $\kappa(A) = \sigma_1(A)/\sigma_d(A)$ is shown to be continuous in $W^{1,q}$ provided the matrix remains nonsingular.
Surface Area: The surface area of the graph of eigenvalue functions is lower semi-continuous.
Compact Self-Adjoint Operators: The results generalize to infinite-dimensional Hilbert spaces. For a Lipschitz family of compact self-adjoint operators, the ordered eigenvalues belong to $C^{0,1}$ and converge in $W^{1,q}$ .

4. Significance and Optimality

Optimality: The authors provide explicit counterexamples (Section 7) demonstrating that their results are sharp.
- They show that continuity fails in the $C^{0,1}$ topology (Lipschitz norm) for the target space.
- They prove that the map is not uniformly continuous in any $W^{1,q}$ or Hölder topology, highlighting the delicate nature of eigenvalue perturbations near points of eigenvalue collision.
Independence of Dimension: Unlike results for polynomial roots (where regularity depends on the degree $d$ ), the eigenvalue stability for matrices holds uniformly regardless of the matrix size $d$ .
Comparison to Polynomials: The results are stronger than those for general polynomials. For hyperbolic polynomials, the solution map is continuous in $W^{1,q}$ , but for general complex polynomials, the regularity is limited to $W^{1,q}$ with $q < d/(d-1)$ . For matrices, the regularity holds for all $q < \infty$ .
Eigenvectors vs. Eigenvalues: The paper emphasizes that while eigenvalues are stable in these Sobolev spaces, eigenvectors generally fail to be even continuous, a well-known phenomenon in perturbation theory.

Conclusion

Parusiński and Rainer establish that the map from a matrix family to its eigenvalues is a robust, continuous operator in Sobolev spaces $W^{1,q}$ for all $q < \infty$ . By utilizing block-diagonalization and the theory of multivalued functions, they resolve the stability problem for both Hermitian and normal matrices, providing a comprehensive framework that applies to singular values, condition numbers, and infinite-dimensional operators. The work clarifies the precise topological limits of this stability, showing that while $W^{1,q}$ continuity holds, the stronger $C^{0,1}$ continuity fails.