Eigenvalues of the Sum of Hermitian Matrices

In the mathematical language of the physical sciences, few objects are as fundamental as Hermitian matrices. They represent the measurable quantities of a system—its energy, momentum, or spin—and their eigenvalues correspond to the only possible outcomes of those measurements. This makes understanding their behavior a cornerstone of fields like quantum mechanics. A central and surprisingly complex question arises when we combine two systems or introduce a new interaction: if we know the eigenvalues of two Hermitian matrices, $A$ and $B$, what can we say about the eigenvalues of their sum, $C = A + B$? The answer is far from a simple addition. It involves a rich interplay of alignment and interference, dictated by elegant mathematical laws.

This article addresses the gap between the naive assumption of simple addition and the complex reality of the eigenvalue sum problem. It provides a comprehensive overview of the principles that govern this fascinating topic and explores their far-reaching consequences.

Across the following chapters, you will embark on a journey from abstract theory to tangible application. In "Principles and Mechanisms," we will uncover the fundamental rules of the game—the inequalities of Hermann Weyl that fence in the possibilities and the principle of eigenvector alignment that allows us to reach those boundaries. Following that, "Applications and Interdisciplinary Connections" will demonstrate how these abstract rules manifest in the real world, from predicting the energy levels of an atom in a magnetic field to designing more stable bridges and robust computational algorithms.

Imagine you are a composer creating a new piece of music by combining two separate melodies. If both melodies are simple and written in the same key and rhythm, their sum is predictable and harmonious. But what if one is a frantic jazz improvisation and the other is a slow, mournful cello line? Their combination is no longer a simple sum. The peaks of one melody might align with the valleys of the other, creating unexpected silences, or they might amplify each other into a cacophony. The result depends entirely on how they are aligned in time.

The world of matrices, particularly the ​​Hermitian matrices​​ that are the bedrock of quantum mechanics, behaves in a remarkably similar way. A Hermitian matrix isn't just an abstract block of numbers; you can think of it as representing a physical observable, like energy or momentum. Its eigenvalues—which are always real numbers—represent the possible values you could measure for that observable. Adding two Hermitian matrices, $A$ and $B$, is like combining two physical systems or introducing a new interaction. For instance, $A$ could be the Hamiltonian (energy operator) of an atom, and $B$ could be the interaction with an external magnetic field. The eigenvalues of the new matrix, $C = A + B$, tell us the new, shifted energy levels of the atom in the field.

You might naively guess that the new eigenvalues are just the sums of the old ones. But, just like with our melodies, this is only true in the special case where the matrices are "compatible" (share the same set of eigenvectors). In the far more interesting general case, the eigenvectors of $A$ and $B$ point in different "directions" in their high-dimensional space. The act of addition becomes a complex interplay of interference and alignment, and a fundamental question arises: given only the eigenvalues of $A$ and $B$, what can we possibly say about the eigenvalues of $A+B$?

Before we can find the exact answer, it helps to put some fences around the field of possibilities. This is the great contribution of the German mathematician Hermann Weyl. His famous inequalities don't give you the exact eigenvalues of the sum, but they provide rigorous upper and lower bounds, telling you where those eigenvalues are allowed to live.

The most fundamental constraint is an exact one, not an inequality. The sum of all eigenvalues of a matrix is its ​​trace​​, and the trace of a sum is the sum of the traces. So, for $C=A+B$, we have:

This is like a conservation law; the total "eigenvalue budget" is fixed. This simple fact is surprisingly powerful. For instance, if you want to maximize the sum of the largest and smallest eigenvalues, you can do so by minimizing the middle one.

Weyl's inequalities then provide further bounds on individual eigenvalues. The most intuitive ones concern the very largest and smallest eigenvalues of the sum. The largest eigenvalue of $A+B$, let's call it $\lambda_{\text{max}}(A+B)$, cannot be larger than the sum of the largest eigenvalues of $A$ and $B$. Similarly, the smallest eigenvalue of the sum is bounded below by the sum of the smallest eigenvalues of $A$ and $B$. For the largest eigenvalue, we have a more detailed bound:

But this is only part of the story. The true magic lies in the "cross-talk" between eigenvalues of different ranks. Weyl's full inequalities reveal a subtle mixing rule. For $n \times n$ matrices, the $k$-th eigenvalue of the sum is bounded by sums of eigenvalues from $A$ and $B$, but the indices must play by a specific rule. For instance, one of the lower bounds for $\lambda_k(A+B)$ is the maximum of all possible sums $\lambda_i(A) + \lambda_j(B)$ where the indices are coupled by the condition $i+j=k+n$ (when eigenvalues are sorted non-increasingly, i.e., $\lambda_1 \ge \lambda_2 \ge \dots \ge \lambda_n$).

Consider two $3 \times 3$ Hermitian matrices, $A$ and $B$. To find a lower bound for the largest eigenvalue of their difference, $\lambda_1(A-B)$, we look at $A+(-B)$. Here, we are interested in the index $k=1$ for the largest eigenvalue, and with $n=3$, the rule becomes $i+j = 1+3=4$. This means the first eigenvalue of the sum is constrained by pairings like the first of $A$ and third of $(-B)$, the second of $A$ and second of $(-B)$, and the third of $A$ and first of $(-B)$. The final lower bound is the maximum of these possible pairings, which in this case turns out to be $2$. These rules prevent a single eigenvalue from running wild; it is tethered to its brethren through these subtle arithmetic relationships.

Weyl's inequalities give us a "safe zone" for the eigenvalues, but can we actually reach the edges of this zone? Can we choose our matrices $A$ and $B$ (while keeping their eigenvalues fixed) to produce a sum $A+B$ whose eigenvalues are precisely the maximum or minimum values allowed? The answer is a resounding yes, and the principle behind it is the geometric idea of ​​eigenvector alignment​​.

Imagine the eigenvectors of $A$ as a set of orthogonal axes, and likewise for $B$. Since we only know the eigenvalues of $B$, not $B$ itself, we are free to rotate its axes relative to $A$'s.

To get the largest possible value for the largest eigenvalue of $A+B$, you would want to align the eigenvector of $A$'s largest eigenvalue with the eigenvector of $B$'s largest eigenvalue. This is "constructive interference"—the two effects amplify each other maximally. For all other eigenvalues, the alignment might be less than perfect.

Conversely, to find the minimum possible value for the largest eigenvalue, the strategy is one of "managed conflict" or "anti-alignment." You don't pair the two largest eigenvalues. Instead, you pair the largest eigenvalue of $A$ with a smaller eigenvalue of $B$, and the second-largest of $A$ with a larger one of $B$, in a way that keeps the resulting maximum value as low as possible. This extremal value is given by a principle established by Alfred Horn and further developed by Vladimir Lidskii. It tells us that the minimum value for the largest eigenvalue, $\lambda_1(A+B)$, is found by considering all possible pairings of eigenvalues from $A$ and $B$. Specifically, for $2 \times 2$ matrices with eigenvalues $\alpha_1 \ge \alpha_2$ and $\beta_1 \ge \beta_2$, the minimum possible value of $\lambda_1(A+B)$ is $\max(\alpha_1+\beta_2, \alpha_2+\beta_1)$. You pair the largest of one with the smallest of the other, and vice-versa, and take the bigger of these two sums. This is the best you can do to sabotage the creation of a large eigenvalue.

This principle of anti-alignment is a powerful tool for optimization. If you have two systems, and you want to combine them while minimizing the sum of the top three resulting energy levels, you would strategically pair the highest energies of one system with the lowest energies of the other. When the eigenvalues of $A$ were $\{12, 10, 8, 6\}$ and those of $B$ were $\{5, 3, 1, -1\}$, the most effective way to minimize the top eigenvalues of the sum was to pair $(12, -1)$, $(10, 1)$, and $(8, 3)$. This kind of thinking is crucial in fields from quantum chemistry to control theory, where one wants to control the spectral properties of combined systems.

So, we see that the eigenvalues of $A+B$ are not a single, fixed set. Instead, for a given set of eigenvalues for $A$ and $B$, there is an entire range of possible eigenvalues for their sum, corresponding to all the possible ways their eigenvectors can be aligned.

The theorems of Weyl and Lidskii, taken together, do something remarkable: they precisely define this range. For any given eigenvalue of the sum, say the second-largest, $\lambda_2(A+B)$, there is a definite minimum and a definite maximum value it can take over all possible alignments. And beautifully, the theory guarantees that this range is a continuous interval—every single value between the minimum and maximum is achievable by some choice of matrices $A$ and $B$.

For example, for specific $3 \times 3$ matrices, a careful application of these principles reveals that the second eigenvalue, $\mu_2$, must lie within the closed interval $[9, 15]$. The lower bound, $9$, is achieved by a particular "anti-alignment" of eigenvectors, while the upper bound, $15$, is achieved by a different, more "constructive" alignment. The existence of this sharp, well-defined range transforms the problem from one of frustrating ambiguity to one of profound and elegant structure.

The journey from a simple question about adding matrices to this rich geometric picture of rotating vectors, eigenvalue permutations, and convex spaces is a perfect example of what makes mathematics so powerful. It takes a problem that seems hopelessly complex on the surface and reveals an underlying order, a set of rules governing a universe of possibilities. The eigenvalues of a sum are not random; they are dancing to a very precise and beautiful choreography.

We have spent a good deal of time wrestling with the mathematical machinery that governs the eigenvalues of Hermitian matrices. We've seen the elegant inequalities of Weyl and the powerful, complete description given by Lidskii's theorem. But as with any deep scientific principle, the question that truly matters is: So what? Where does this abstract dance of numbers meet the messiness and complexity of the real world? The answer, it turns out, is everywhere. The principles we’ve uncovered are not merely mathematical curiosities; they are fundamental constraints on how the universe works, from the smallest particles to the largest engineered structures. This chapter is a journey into that world, to see these abstract rules in action.

Perhaps the most natural and profound home for Hermitian matrices is quantum mechanics. In that strange and wonderful realm, every physical quantity you can measure—energy, momentum, spin—is represented by a Hermitian operator. And what are its eigenvalues? They are the only possible outcomes of a measurement. The "spectrum" of an operator is not just a collection of numbers; it is the very set of realities that an experiment can reveal. An atom's energy levels, for instance, are the eigenvalues of its Hamiltonian operator.

Now, imagine we have a simple system, like a hydrogen atom floating in empty space. We know its Hamiltonian, $H_0$, and we have painstakingly calculated or measured its energy levels, which are the eigenvalues of $H_0$. Now, what happens if we place this atom in an external magnetic field? The field adds a new piece to the total energy, a new Hermitian operator, let's call it $V$. The new Hamiltonian for the atom in the field is now $H = H_0 + V$.

Suddenly, we are faced with precisely the problem we have been studying: what are the eigenvalues of a sum of two Hermitian matrices? Must we solve the whole, horrendously complicated problem all over again? The remarkable answer is no, not entirely. The Weyl inequalities come to our rescue, giving us immediate, powerful bounds on the new energy levels of the perturbed atom, using only our knowledge of the original atom and the field itself. We can predict how the energy spectrum will shift and spread without solving a single new differential equation. This is an immense labor-saving device, but more importantly, it is a tool for thought. It allows a physicist to build an intuition for how systems respond to change.

In some idealized scenarios, such as when the energy levels of the original system and the perturbation both form simple arithmetic progressions, these bounds become astonishingly precise. They can pinpoint the exact minimum or maximum possible value for a specific energy level in the combined system. While nature is rarely so perfectly ordered, these "toy models" are invaluable; they reveal the underlying structure of the problem in its purest form and show just how much predictive power is packed into these theorems.

Beyond the quantum world, the lessons of eigenvalue sums are central to the art of design and optimization. The general problem is this: you have two components, $A$ and $B$, each with a known set of intrinsic properties (their eigenvalues). You can combine them in different ways (by changing the relative orientation of their eigenvectors). What is the best, or worst, possible outcome for the combined system, $A+B$?

Think of designing a bridge. The stiffness matrix of the structure, which is Hermitian, has eigenvalues related to the squares of its natural vibration frequencies. A small lowest eigenvalue means a low fundamental frequency, making the bridge susceptible to swaying in the wind or from foot traffic. Let's say we have the main structure ($A$) and a set of reinforcing beams ($B$). Both have known structural properties. The question is: how can we add the reinforcements to the main structure to create the most stable bridge? This often translates to maximizing the smallest eigenvalue of the combined stiffness matrix, $A+B$. This is no longer a hypothetical question. It's a problem of safety and engineering. The tools we have developed allow us to find the absolute maximum possible value for that lowest frequency, giving us a hard ceiling on how stable we can possibly make our design.

This principle applies to countless scenarios. We might want to minimize the financial risk (an eigenvalue-related measure) of a combined portfolio, or maximize the signal-to-noise ratio in a communication system. Sometimes, the goal is to optimize a more complex quantity, like the determinant of a system's matrix, which might relate to its volume or information content. Or perhaps we need to minimize a quantity like the trace norm, which can represent the total energy or cost of a system. In all these cases, the logic is the same: Lidskii's theorem defines the "feasible region"—the complete set of possible eigenvalue vectors for $A+B$. Our job as designers is to navigate this space to find the point that optimizes our desired objective function.

Sometimes, the problem is one of tuning. Imagine you have a system $A$ and you are adding an interaction $B$ whose strength you can control via a parameter $b$. For each value of $b$, there is a range of possible outcomes for the eigenvalues of $A+B$. One might ask: what is the best we can do to stabilize the system, and what is the optimal strength $b$ to achieve this? A fascinating case arises when we seek to minimize the largest eigenvalue (the "peak response") of the combined system. We might find that there is a specific, optimal strength $b_0$ where the eigenvalues of the two parts "interfere" in just the right way to produce the most stable configuration. This is the mathematical echo of phenomena like resonance and anti-resonance that appear all over physics and engineering.

Of course, the real world rarely gives us a blank check. More often than not, optimization comes with constraints. An engineer might need to design a circuit that maximizes performance while keeping its total power consumption (perhaps related to the determinant) fixed. In this scenario, we are not free to roam the entire feasible region of eigenvalues. We must stick to a path defined by our constraint. The problem then becomes one of finding the best point along this specific path. This beautiful interplay between fundamental mathematical law and practical, ad-hoc constraints is the daily reality of science and engineering.

So far, we have focused on combining systems. But there is a related, equally important question: if you have a single system, how sensitive is it to being disturbed? If you change a matrix $A$ just a little bit, to a new matrix $B$, how much can its eigenvalues—its fundamental properties—change? This is the domain of perturbation theory, and it is vital for understanding the stability and robustness of any system.

Think of a computer calculating the energy levels of a molecule. Due to finite precision arithmetic, the matrix it actually uses isn't the true Hamiltonian $A$, but a slightly different one, $B$. We need to know if this tiny error could lead to a wildly different, and thus meaningless, result.

The Hoffman-Wielandt theorem provides a profound and deeply reassuring answer for this question. It gives a precise upper bound on how much the set of eigenvalues can shift. It tells us that the "distance" between the spectrum of $A$ and the spectrum of $B$ (measured by the sum of squared differences of corresponding eigenvalues) is no larger than the "distance" between the matrices $A$ and $B$ themselves (measured by the Frobenius norm, which is just the sum of the squared magnitudes of their element-wise difference).

In simpler terms: small changes to the matrix lead to small changes in the eigenvalues. The spectrum is not "chaotic"; it is stable. This guarantee is the bedrock upon which much of numerical linear algebra is built. It tells us that our computer simulations are not a house of cards, ready to collapse from the slightest rounding error. It also tells a physicist that the energy levels of an atom will not change catastrophically if it is subjected to a tiny, stray electric field. It is a mathematical certificate of the robustness of the physical world.

Looking back, it is truly remarkable. We started with a simple, abstract question about adding numbers in a grid. We have ended with insights into quantum mechanics, structural engineering, optimization, control theory, and the stability of computation. The same set of mathematical principles—the same elegant inequalities—are found at work in all these disparate fields. It is a stunning demonstration of the unity of science. The spectrum of a Hermitian matrix is more than a list of numbers; it is a fingerprint of a system, and the theorems we have explored are the universal rules that govern how these fingerprints can, and cannot, change when systems interact.