The Theorem an AI Helped Prove: When Matrices Refuse to Go Negative

Every number in a matrix has a sign — positive or negative — and for decades nobody knew whether one particular family of matrices, born from problems in control theory, could ever be trusted to keep those signs all in one direction. The answer, it turns out, was hiding in an equation that engineers have been solving happily for a century, and it took an unusual collaborator to finally pin it down: artificial intelligence.

The puzzle began as a rumor. It circulated "largely through word of mouth among the Italian controls community," the paper notes, seemingly invisible to the rest of the mathematical world. A retired control theorist in Catania, Luigi Fortuna, had apparently suspected something elegant: when you solve a certain classical matrix equation with a matrix in a special "companion" form, the answer might always be composed entirely of nonnegative entries — never a negative number anywhere. A recent paper by Ferrante (2026) finally wrote the conjecture down in print and proved it for a restricted case)Skip — but only when every eigenvalue of the matrix was real. Whether the claim held when eigenvalues came in complex pairs, the common situation, remained open.

Now three researchers — Miaomiao Wang, Patrizio Colaneri of the Polytechnic University of Milan, and Jie Chen of City University of Hong Kong — have settled it, and they did so with a proof partly generated by an AI coding assistant. The result, to appear in the journal Automatica, closes the conjecture completely and reveals a sharp and surprising asymmetry between the continuous-time and discrete-time versions of the problem. One is beautifully true; the other is starkly, and immediately, false.

The Science

To understand what was proved, it helps to meet the equation. The Lyapunov equation, named for the Russian mathematician Aleksandr Lyapunov, is a workhorse of systems theory:

$$XA+A^{\mathsf{T}}X=-Q.$$

Here $A$ is the "state matrix" describing how a system evolves, $Q$ is a symmetric forcing matrix that is positive semidefinite (written $Q⪰0$, meaning all its eigenvalues are nonnegative), and $X$ is the unknown, the solution we seek. Long ago, Lyapunov showed that if $A$ is Hurwitz — every eigenvalue having negative real part, so the system decays to equilibrium rather than blowing up — then the equation has exactly one solution $X$, and that solution is itself positive semidefinite.

But note what that guarantees and what it does not. Positive semidefiniteness is a statement about the global shape of a matrix, its eigenvalues. It says nothing about the individual entries. A matrix can be positive semidefinite while still containing negative numbers scattered among its cells — indeed, this is completely ordinary. The conjecture was far stronger: that under the right conditions, every single entry of $X$ would be nonnegativemars. Not just the matrix as a whole, but each of its $n^2$ numbers.

The twist is what "the right conditions" means. The state matrix $A$ must be a companion matrix, a special form that encodes a polynomial in its bottom row and ones along its superdiagonal:

$$A=\left[\begin{array}{cccc}0& 1& & 0\\ & \ddots & \ddots & \\ 0& & 0& 1\\ -{\sigma }_n& -{\sigma }_{n-1}& \cdots & -{\sigma }_1\end{array}\right].$$

Companion matrices arise constantly: the roots of the polynomial are exactly the eigenvalues of the matrix, so they are a natural way to "linearize" a scalar polynomial into a matrix problem. The conjecture says that when $A$ is a Hurwitz companion matrixboarding, the solution $X$ is entrywise nonnegative for every positive semidefinite forcing $Q$.

The earlier proof by Ferrante relied heavily on Vandermonde matrices — matrices that diagonalize a companion matrix when its eigenvalues are all distinct — and on a positivity property of certain "Cauchy-like" matrices. That approach collapsed when eigenvalues were allowed to be complex and repeated, because the diagonalizing structure degenerates.

The new proof sidesteps all of that. "Unlike in Ferrante," the authors write, the proof "has no reference to Vandermonde matrices." Instead, it uses only the bare structure of the companion form plus the one fact everyone already knew — that the Lyapunov solution exists and is unique. The key technical machinery is a family of Horner polynomials, named for the same Horner who invented the efficient scheme for evaluating polynomials:

$$h_p(s)=s^p+{\sigma }_1{s}^{p-1}+\cdots +{\sigma }_p,h_0(s)=1.$$

These polynomials have a magical relationship with companion matrices: applying $h_p(A)$ to the last standard basis vector simply shifts it to another basis vector. That single observation, the paper shows, unlocks everything.

What They Found

The main theorem is clean. For any real Hurwitz companion matrix $A$ and any symmetric $Q⪰0$, the unique symmetric solution $X$ of the continuous-time Lyapunov equation satisfies

$$X_{k\ell }\ge 0,1\le k,\ell \le n.$$

Every entry, every one of the $n^2$ numbers, is nonnegative. And in a strict Version — Corollary 2.2 — if $Q$ is positive definite rather than merely semidefinite, then every entry of $X$ is strictly positive: $X_{k\ell }>0$.

The proof architecture is subtle and worth pausing on, because it reveals how the companion structure does the heavy lifting. The authors first prove a "mixed accretivity" lemma about the coefficients of adjugate polynomials — built up by induction on matrix dimension — then transfer that positivity into the Lyapunov setting through a controllability Gramian and the Horner basis. The upshot is that a certain inner-product expression $X_{k\ell }$ is revealed to be the trace of a product of positive semidefinite matrices, which is necessarily nonnegative.

But the most striking finding is the sharp contrast with the discrete-time analogue. The discrete-time Lyapunov equation, also called the Stein equation, reads

$$X-A^{\mathsf{T}}XA=Q,$$

where now $A$ is Schur stable (all eigenvalues inside the unit disk) rather than Hurwitz. One might naïvely expect the same positivity miracle. It fails spectacularly.

The counterexamples are tiny. In dimension two — the smallest dimension where failure is possible, since dimension one always works — the authors exhibit a perfectly respectable Schur companion matrix built from the polynomial $a_2(z)=z^2+\frac{1}{2}z-\frac{1}{4}$. Even with the identity forcing $Q=I$, the solution comes out as

$$X_2=\frac{1}{25}\left[\begin{array}{cc}31& -16\\ -16& 96\end{array}\right],$$

whose $(1,2)$ entry is $-16/25$, stubbornly negative.

One might suspect the failure is an artifact of having a negative coefficient in the polynomial. So the authors construct a second example — a cubic whose coefficients are all strictly positive: $a_3(z)=z^3+\frac{1}{10}{z}^2+\frac{1}{5}z+\frac{3}{5}$. It is strictly Schur stable (proved neatly with Rouché's theorem), and yet the Stein solution still contains a negative off-diagonal entry, $-36/475$. Even with identity forcinggged, even with all-positive coefficients, the discrete-time world refuses to cooperate.

Why This Changes Things

The result closes an old mathematical question, but its wider significance is twofold: it reveals something structural about companion matrices, and it may well be a landmark in how mathematics gets done.

First, the mathematics. The fact that the continuous-time theorem holds with no assumptions on the spectrum — no requirement that eigenvalues be real, distinct, or anything else — was the entire gap, and the new proof leaps it in one stroke, holding for matrices with complex eigenvalue pairs and repeated roots alike. Moreover, the theorem is coordinate-dependent: the entrywise positivity holds specifically in the controller companion coordinates, and is not invariant under similarity transforms. That is not a weakness but a feature — it tells you exactly where the structure lives, and that the companion form is doing real work, not just dressing up a more general fact.

The authors are candid about scope. "We concede that while the conjecture and its proof are remotely related to systems theory subjects," they write, "they are essentially of academic interest and are driven by intellectual curiosity. We do not purport that the problem will be of significant practical interest." That honesty is refreshing, and it is worth taking seriously rather than dismissing: pure mathematics is often justified precisely by the fact that we cannot predict which elegant structural facts will later become load-bearing in applications. The Lyapunov equation underpins stability analysis, optimal control, and model reduction; knowing the sign patterns of its solutions for companion matrices is the kind of fact that quietly informs how engineers reason about the systems they design.

There is also a lesson embedded in the discrete-time failure. The two equations look nearly identical, differing only in a minus sign and a shift of the stability region from the left half-plane to the unit disk. Yet one obeys a beautiful entrywise positivity law and the other resists it at the very first nontrivial dimension. That asymmetry is itself a kind of finding — a reminder that superficially parallel mathematical structures can diverge sharply, and that intuition built on one must be re-examined for the other.

Second, and perhaps most consequential for the field at large: this appears to be one of the first published AI-assisted theoretical results in control theory. The proof, the authors state, is "partly due to CODEX-5.6 Sol Ultra, and partly due to extensive, informed trainings provided by the authors." They go further, calling it "potentially the birth of AI in control systems theory." The honest accounting is important: the AI did not march in and solve the problem alone; the authors identified the right constructs — the Horner basis, the adjugate accretivity, the dimension reduction — and guided the collaboration. But the tools helped discover and verify an elementary proof of something that had resisted a community's effort.

This matters beyond this one theorem. Mathematics has long prized proofs that are elegant and "elementary" — built from the minimum possible machinery. The new proof is exactly that: "the rest of the proof is solely algebraic manipulation," the authors note, and it uses only the companion structure plus the existence and uniqueness of the Lyapunov solution. Discovering such minimal proofs is often the hardest part of mathematics, and here is a case where the discovery process involved a machine doing substantial creative work under human direction.

What's Next

The paper leaves a clear open thread dangleiting in the discrete-time case. Having shown that entrywise nonnegativity generally fails for the Stein equation, the authors characterize when it can be recovered: after a Cayley transform, which maps the unit disk to the left half-plane, and a change of basis, the solution becomes entrywise nonnegative again. But this recovery happens in a transformed coordinate system, "a transformed companion basis that depends only on the state matrix" — not the original coordinates.

The open direction the authors flag is precise: "to characterize the Schur companion matrices and positive-semidefinite forcing matrices for which the Stein solution is entrywise nonnegative in the original coordinates." In other words, the discrete failure is proved, but its full geography — exactly which cases do and do not fail — remains unmapped.

Beyond this specific question looms a larger one about the role of AI in mathematical discovery. If this paper is truly the watershed it claims to be, then the coming years will see many more theorems where machines contribute to proofs. The hard questions — how much credit to assign, how to verify AI-assisted arguments, what happens to mathematical intuition when proofs outpace human comprehension — are just beginning to be asked. This result offers a hopeful first data point: a collaboration in which AI and humans together produced something elementary, verifiable, and true, in a field where progress had stalled.

For the reader, the takeaway is quietly profound. The Lyapunov equation, taught to every engineering student, has been fully understood for over a century. And yet, buried inside it, was a hidden law about the signs of individual numbers that took until now — and, fittingly, an AI collaborator — to uncover. It is a reminder that even in the most settled mathematics, unexamined structure waits, and that the very tools we are building to automate reasoning may be the ones to finally find it.