Weyl's inequalities

In many scientific domains, from quantum mechanics to structural engineering, a central challenge is to predict the properties of a system when it is combined with or perturbed by another. If the properties of each component are known—for instance, the energy levels of two separate atoms—what can we say about the energy levels of the combined system? The answer is often not straightforward, as the interaction itself introduces a layer of complexity. This creates a significant knowledge gap: without knowing the exact nature of the interaction, can we say anything definitive about the outcome?

This is precisely the problem addressed by Weyl's inequalities, a cornerstone theorem of linear algebra concerning the eigenvalues of Hermitian matrices. These inequalities provide a powerful solution by establishing rigorous, predictable bounds on the eigenvalues of a sum of matrices, even when the exact result is unknowable. They transform uncertainty into a defined range of possibilities.

This article explores the power and elegance of Weyl's inequalities. The first chapter, ​​Principles and Mechanisms​​, will unpack the mathematical foundation of the inequalities, demonstrating how they pin down the eigenvalue spectrum of a matrix sum and provide a guarantee of stability under perturbations. Following this, the chapter on ​​Applications and Interdisciplinary Connections​​ will reveal the far-reaching impact of this theorem, showing how it underpins our understanding of stability in quantum systems, ensures the reliability of engineering designs, and validates results in modern computational science.

Imagine you have two separate collections of musical tuning forks. For each collection, you know the exact set of frequencies they produce when struck—let’s call these the frequency “spectrums.” Now, what if you were to create a new, combined system by somehow coupling these two sets of tuning forks together? Could you predict the new spectrum of frequencies? It seems like a hard problem. The new frequencies will surely depend on how you connect them, not just on the original frequencies. You might guess that you can’t know the new frequencies exactly. And you’d be right. But what if I told you that you could, with absolute certainty, determine a precise range in which each new frequency must lie?

This is the very essence of the problem that Hermann Weyl solved for a class of mathematical objects called ​​Hermitian matrices​​. In the quantum world, these matrices represent physical observables like energy, momentum, or spin. Their eigenvalues are the possible values those quantities can take—the allowed energy levels of an atom, for instance. So, understanding how eigenvalues behave when we add matrices is like understanding how energy levels shift when two physical systems are combined. Weyl's inequalities give us the rules of this combination.

Let's get to the heart of the matter. Suppose we have two $n \times n$ Hermitian matrices, $A$ and $B$. We know all their eigenvalues, which we'll list in non-decreasing order:

• Eigenvalues of $A$: $\lambda_1(A) \le \lambda_2(A) \le \dots \le \lambda_n(A)$
• Eigenvalues of $B$: $\lambda_1(B) \le \lambda_2(B) \le \dots \le \lambda_n(B)$

We are interested in the eigenvalues of their sum, $C=A+B$, which we'll call $\lambda_k(C)$. Weyl discovered that every single eigenvalue $\lambda_k(C)$ is trapped in a specific interval, defined by two beautiful inequalities.

For the ​​lower bound​​, which tells us the smallest possible value for $\lambda_k(C)$:

And for the ​​upper bound​​, which tells us the largest possible value:

At first glance, these formulas might seem a bit dense, but the idea is wonderfully intuitive. To find the floor for the $k$-th eigenvalue, you look at all the ways you can "build" the index $k$ (as $i+j-1=k$) by pairing eigenvalues from $A$ and $B$, and you take the most optimistic pairing. To find the ceiling, you do a similar search, but the pairing rule ($i+j=k+n$) is different.

Let’s see this in action. Suppose we have two $3 \times 3$ Hermitian matrices, $A$ and $B$. The eigenvalues of $A$ are $\{8, 9, 10\}$ and the eigenvalues of $B$ are $\{-6, -6, 12\}$. We want to find the possible range for the second eigenvalue, $\lambda_2(A+B)$. Here, $n=3$ and $k=2$.

First, the lower bound. We need $i+j = k+1 = 3$. The possible pairs of indices $(i, j)$ are $(1, 2)$ and $(2, 1)$.

• For $(1, 2)$: $\lambda_1(A) + \lambda_2(B) = 8 + (-6) = 2$.
• For $(2, 1)$: $\lambda_2(A) + \lambda_1(B) = 9 + (-6) = 3$. The lower bound is the maximum of these values, so $\lambda_2(A+B) \ge 3$.

Now, the upper bound. We need $i+j = k+n = 2+3=5$. The possible pairs are $(2, 3)$ and $(3, 2)$.

• For $(2, 3)$: $\lambda_2(A) + \lambda_3(B) = 9 + 12 = 21$.
• For $(3, 2)$: $\lambda_3(A) + \lambda_2(B) = 10 + (-6) = 4$. The upper bound is the minimum of these values, so $\lambda_2(A+B) \le 4$.

And there it is! Like a detective pinning down a suspect's location, we’ve determined that the second eigenvalue of the combined system must lie in the interval $[3, 4]$. We don't know the exact value without knowing the matrices themselves, but we've constrained it to a remarkably small window. This interval, the difference between the upper and lower bounds, is a fundamental measure of the uncertainty that arises from not knowing the alignment of the matrices' eigenvectors.

You might be thinking, "This is nice, but can the bounds ever be exact?" The answer is a resounding yes, and when they are, it reveals something deep about the system's structure.

Consider a special case: a $2 \times 2$ matrix $A$ whose eigenvalues are both $5$. In the world of Hermitian matrices, the only way this can happen is if $A$ is the matrix $\begin{pmatrix} 5 & 0 \\ 0 & 5 \end{pmatrix}$, which we can write as $5I$, where $I$ is the identity matrix. This matrix is special; it doesn’t rotate or shear vectors, it just scales them all by a factor of 5.

Now, let's take another matrix $B$ with eigenvalues $1$ and $3$, and form the sum $A+B = 5I+B$. What are the eigenvalues of this new matrix? Since adding $5I$ just shifts everything, the action of $A+B$ on an eigenvector $v$ of $B$ is:

The new eigenvalues are simply the eigenvalues of $B$, each increased by 5! So the eigenvalues of $A+B$ must be exactly $1+5=6$ and $3+5=8$. The largest eigenvalue is precisely 8.

Let's see what the inequalities we just learned tell us. We want the bounds for the largest eigenvalue, so $k=n=2$.

• For the lower bound, we need $i+j=k+1=3$. The possible pairs are $(1,2)$ and $(2,1)$. We get $\lambda_1(A)+\lambda_2(B)=5+3=8$ and $\lambda_2(A)+\lambda_1(B)=5+1=6$. The maximum is 8, so $\lambda_2(A+B) \ge 8$.
• For the upper bound, we need $i+j=k+n=4$. The only pair is $(2,2)$. We get $\lambda_2(A)+\lambda_2(B)=5+3=8$. The minimum is 8, so $\lambda_2(A+B) \le 8$. Combining these, the inequalities predict that $8 \le \lambda_2(A+B) \le 8$, meaning the largest eigenvalue must be exactly 8! This matches our direct calculation. The upper and lower bounds coincide because one of the matrices, $A$, is a scalar multiple of the identity, meaning its eigenvectors can align perfectly with the eigenvectors of $B$. Weyl's inequalities are "tight"—you can't construct a narrower range that would be true for all possible matrices, and in this special case, they collapse to a single point.

Perhaps the most profound application of Weyl's inequalities is in the study of ​​perturbations​​. In the real world, our models are never perfect. We might have a perfect theoretical model of a system (an atom, a bridge, a planetary orbit), described by a matrix $A$. But in reality, there are always tiny, unaccounted-for influences—a stray magnetic field, a gust of wind, the gravitational pull of a passing asteroid. We can lump all these small effects into a "perturbation" matrix, $E$. The real system is then described by $A+E$.

A crucial question for any physicist or engineer is: if the perturbation $E$ is small, will the change in the outcome (the eigenvalues) also be small? If a tiny disturbance could cause a catastrophic change in the system's behavior, our models would be useless. We need stability.

Weyl's inequality provides the ultimate guarantee of this stability. Let's say we can quantify the "size" of the perturbation $E$ by its ​​spectral norm​​, $\|E\|$, which is the largest absolute value of its eigenvalues. Let's call this size $\epsilon$. This means all eigenvalues of $E$ are contained in the interval $[-\epsilon, \epsilon]$.

Now we apply Weyl's inequalities to the sum $A+E$. Let $\alpha_k$ be the eigenvalues of $A$ and $\beta_k$ be the eigenvalues of the perturbed system $A+E$. The inequalities tell us:

Since $\lambda_{\min}(E) \ge -\epsilon$ and $\lambda_{\max}(E) \le \epsilon$, we get:

This can be rewritten in a wonderfully simple and powerful form:

This is a beautiful result. It states that the shift in any eigenvalue is no larger than the size of the perturbation. A small cause leads to a small effect. The energy levels of an atom won't scatter randomly if it enters a weak electric field. The fundamental frequencies of a violin string won't change dramatically if the temperature shifts slightly. This mathematical certificate of stability is what allows us to build reliable models of the physical world.

The power of Weyl's inequalities doesn't stop with simple sums. They provide a whole toolkit for understanding how eigenvalues transform.

• ​​Subtraction?​​ What about the eigenvalues of $A-B$? Just think of it as $A+(-B)$. The eigenvalues of $-B$ are simply the negatives of the eigenvalues of $B$, and you can apply the inequalities as before.
• ​​Scaling?​​ What about $A+2B$? Easy. The eigenvalues of $2B$ are just twice the eigenvalues of $B$. The same logic applies.
• ​​More than two matrices?​​ What about $A+B+C$? You can apply the inequalities iteratively. First, find the possible range for the eigenvalues of an intermediate matrix $D=A+B$. Then, using that range of possibilities, apply the inequalities again to find the bounds for $D+C$. The method is robust and extendable.

Weyl's inequalities open a window into the hidden structure of linear algebra. They transform a seemingly impossible problem—predicting the exact eigenvalues of a sum—into a tractable one: finding hard boundaries on those values. They show us that while we may not know everything about a combined system, we are far from knowing nothing. And in science and engineering, knowing the bounds of possibility is often all the power we need.

Now that we have grappled with the mathematical bones of Weyl's inequalities, let us dress them in flesh and blood. You might be tempted to see these inequalities as a dry, abstract piece of linear algebra—a curiosity for the pure mathematician. But nothing could be further from the truth! This is where the magic truly begins. Like a master key, Weyl’s inequalities unlock doors in a surprising array of fields, from the subatomic realm of quantum mechanics to the practical world of engineering and computer science. The common thread is a single, profound question: what happens to a system when you give it a little nudge?

Imagine a perfectly balanced, isolated system. In physics, this might be a hydrogen atom, floating alone in space. In engineering, it could be a bridge, standing still in calm weather. We can often describe the essential properties of such systems with a Hermitian matrix—let’s call it $A$—whose eigenvalues represent crucial physical quantities: the discrete energy levels of the atom, the natural vibration frequencies of the bridge, and so on.

But the real world is never perfect. The atom is bathed in a weak electric field; a gust of wind pushes on the bridge. We have introduced a perturbation, a small change that we can represent by another Hermitian matrix, $E$. The new system is described by the sum, $A+E$. The vital question is: what are the new energy levels, the new vibrational frequencies? Do they change a little, or a lot? Can the system become unstable?

This is the essence of perturbation theory, a cornerstone of modern physics and engineering, and Weyl's inequalities provide the first, most fundamental answer. They give us a rock-solid guarantee. They tell us that the new eigenvalues of $A+E$ cannot stray too far from the old ones. Specifically, the simplest form of the inequality tells us that the $k$-th eigenvalue of the perturbed system is trapped in a predictable interval:

$\lambda_k(A) + \lambda_{\min}(E) \le \lambda_k(A+E) \le \lambda_k(A) + \lambda_{\max}(E)$

Think about what this means. If our perturbation $E$ is "small"—meaning its eigenvalues are all close to zero—then every single eigenvalue of the new system $A+E$ must remain close to its original counterpart in $A$. A small nudge results in a small change. The inequalities provide a rigorous, quantitative bound on this change. For a quantum system, this means the energy levels shift slightly but don't suddenly fly off to infinity. For a bridge, the resonant frequencies are altered, but in a controlled way. The stability of the world, in many ways, is underwritten by this elegant mathematical fact.

This same principle extends to the world inside our computers. When we ask a machine to calculate the eigenvalues of a matrix $A$, it never gets the answer perfectly right due to finite precision and rounding errors. What it actually calculates are the eigenvalues of a slightly different matrix, $A+E$, where $E$ is the matrix of tiny computational "noise." How can we trust the result? Weyl's inequality comes to the rescue! If we can put a bound on the size of the error—for instance, by knowing the maximum possible magnitude of any entry in $E$, which in turn bounds its spectral norm and thus its eigenvalues—we can establish a guaranteed window of accuracy for the computed eigenvalues. Without this, much of modern scientific computation, from climate modeling to aircraft design, would be built on sand.

The power of Weyl’s inequalities goes far beyond simply bounding the shift. They reveal a rich, interwoven structure linking the entire spectrum of $A$ to the spectrum of $A+B$. It’s not just a relationship between corresponding eigenvalues; it's a web of connections.

For example, the inequalities in their more general form, like $\lambda_{i+j-1}(A+B) \le \lambda_i(A)+\lambda_j(B)$, give us a whole family of bounds. For any given eigenvalue of the sum, say $\lambda_2(A+B)$, there might be multiple ways to combine eigenvalues from $A$ and $B$ to create an upper bound. Nature—or rather, mathematics—demands that the tightest of these bounds is the one that holds true. This reveals a subtle interplay; the effect of a perturbation on one eigenvalue is constrained not just by one, but by a whole committee of other eigenvalues.

This deeper understanding allows us to ask more sophisticated questions. Instead of just asking, "By how much does the third eigenvalue change?", we can ask something far more practical: "Given a system $A$ and a set of possible perturbations $B$, can we guarantee that at least one of its resonant frequencies will rise above a critical threshold?" This might be crucial for determining if a circuit will begin to oscillate, or if a structure will fail. Weyl's inequalities provide the tools to answer just such a question, by allowing us to calculate the absolute minimum value that, say, the largest eigenvalue of the system must have, no matter how the perturbation is specifically configured.

Sometimes, our perturbation isn't a random, fuzzy cloud of noise. It's a sharp, targeted change. Imagine you have a complex network, and you add just one new connection. Or in a machine learning model, you update your weights based on a single new piece of data. These kinds of modifications are often represented by adding a low-rank matrix, most simply a rank-one matrix.

Weyl's inequalities are magnificently adapted to this scenario as well. A rank-one matrix has only one non-zero eigenvalue. For a perturbation matrix $B$ with eigenvalues, say, $\{-3, 0, 0\}$, the inequalities tell us exactly how this one influential value ripples through the spectrum of the original matrix $A$. We can determine the tightest possible bounds on the resulting eigenvalues of $A+B$. This gives us incredible insight into how simple, targeted changes affect a complex system, a principle that is fundamental to iterative optimization algorithms and control theory.

Perhaps the most beautiful aspect of a great scientific principle is not just what it explains, but how it points toward something deeper. Weyl’s inequality is a perfect example. We started with the sum of two matrices, $A+B$. But what about three? Or four?

One might guess that a similar rule holds, and one would be right. By a wonderfully simple and elegant trick, we can derive the rule for three matrices from the rule for two. We just group them: think of $A+B+C$ as $(A+B) + C$. We can apply Weyl's inequality to this grouping. First, we treat $(A+B)$ as a single entity and get a bound involving its eigenvalues and those of $C$. Then, we apply the inequality again to the eigenvalues of $(A+B)$ to break them down in terms of $A$ and $B$.

When you follow this logic through, a stunningly simple pattern emerges. The inequality for two matrices can be written as $\lambda_k(A+B) \le \lambda_i(A) + \lambda_j(B)$, given the indices satisfy $i+j=k+1$. When we extend this to three matrices, $A, B,$ and $C$, the process of repeated application reveals that the corresponding inequality holds when the indices satisfy $i+j+l = k+2$. Do you see the pattern? For a sum of $m$ matrices, the condition becomes a sum of $m$ indices equaling $k + (m-1)$.

This is more than just a formula; it's a glimpse into the deep, recursive structure of mathematics. A simple, powerful rule, when applied to itself, builds a more complex but equally elegant rule, like a set of Russian nesting dolls. It shows us that the relationship between matrices and their eigenvalues is not an arbitrary mess, but a landscape governed by profound and beautiful ordering principles. From the jitters of a quantum particle to the stability of a giant bridge, and into the very heart of abstract mathematical structure, Weyl's inequalities provide a constant, reliable, and deeply insightful guide.