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Elementwise Positivity of the Solution to Lyapunov Equation for Hurwitz Companion Matrices

Miaomiao Wang, Patrizio Colaneri, Jie Chen

arXiv:2609.11651v1eess.SYcs.SY

TL;DR

The paper addresses whether continuous-time Lyapunov solutions for Hurwitz companion matrices are entrywise nonnegative under positive-semidefinite forcing. It proves the conjecture without a real-spectrum assumption using Horner polynomial matrices, while showing that the direct discrete-time analogue fails and that positivity can be recovered in a transformed companion basis.

  • Problem

    For companion state matrices, prior work had not established entrywise nonnegativity of the continuous-time Lyapunov solution under arbitrary positive-semidefinite forcing without restricting the spectrum to real eigenvalues.

    Source: §1 · p. 1

  • Method

    The proof uses Horner polynomial matrices and a Cayley transformation with a change of basis for the discrete-time analysis.

    Source: §1 · p. 1, §6 · pp. 5–6

  • Results

    The continuous-time conjecture holds without a real-spectrum assumption, whereas the direct discrete-time analogue fails even with identity forcing and positive polynomial coefficients.

    Source: §1 · pp. 1–2, §7 · p. 6

  • Takeaways & Limitations

    Entrywise positivity is established for continuous-time solutions in controller companion coordinates and for a congruence-transformed discrete-time solution, not the original discrete coordinates.

    Source: §7 · p. 6, §6 · p. 6

    The authors describe the work as primarily academic and do not claim significant practical interest; the transformed discrete-time result does not imply entrywise nonnegativity in the original coordinates.

    Source: §1 · p. 2, §6 · p. 6

Abstract

from arXiv · show

We prove, with the aid of AI, that for every real symmetric forcing matrix $Q\succeq0$, the unique solution of a continuous-time Lyapunov equation is entrywise nonnegative whenever the state matrix is a real Hurwitz companion matrix. This proves an earlier conjecture. The proof makes no assumption on the spectrum of the state matrix, and is made possible by means of using Horner polynomial matrices. We also show that discrete-time counterpart of the conjecture is false: Counterexamples can be readily constructed to show that Schur companion matrices, including one generated by a polynomial with all positive coefficients, yield solutions with negative off-diagonal entries even with identity forcing.

1. Introduction

The paper completes a conjecture on entrywise nonnegativity for continuous-time Lyapunov solutions with Hurwitz companion state matrices, then examines the discrete-time analogue. Its proof covers matrices with complex eigenvalues, while counterexamples show the direct discrete-time claim fails.

1. Introduction

  • Earlier structural solution methods for companion-matrix equations did not address whether all solution entries are nonnegative under arbitrary positive-semidefinite forcing.The conjecture was previously proved only under a real-spectrum assumption using Cauchy-like matrix positivity (Ferrante, 2026b).Source: §1 · p. 1
  • The paper proves the conjectured entrywise nonnegativity result without assuming a real spectrum, using Horner matrix polynomials and adjugate polynomial accretivity.The proof uses the companion structure and the unique semidefinite solution, without Vandermonde matrices.Source: §1 · p. 1
  • The discrete-time analogue fails for Schur companion matrices, including a two-dimensional example with identity forcing and another with all-positive polynomial coefficients.The solution remains positive semidefinite as an observability Gramian, but can have negative off-diagonal entries.Source: §1 · pp. 1–2
  • The authors characterize the work as primarily academic and curiosity-driven, and do not claim significant practical interest.They describe its connection to structured realizations and positive systems as remote.Source: §1 · p. 2
  • The proof is presented as a joint effort of the authors and AI tools, including informed training, and described as a first AI-assisted theoretical result in open controls literature to the authors’ knowledge.The authors qualify the novelty claim with “to the best of our knowledge.”Source: §1 · p. 2

2. Continuous-time problem and main result

This section considers the continuous-time Lyapunov equation for a real Hurwitz controller companion matrix and symmetric positive-semidefinite forcing. It states an entrywise conclusion and notes that its coordinate-specific scope allows nonreal eigenvalues.

2. Continuous-time problem and main result

  • Theorem 2.1 states an entrywise conclusion for the unique symmetric solution when the real controller companion matrix is Hurwitz and Q is positive semidefinite.The equation’s unique solution has an integral representation that is positive semidefinite in the Loewner order.Source: §2 · p. 2
  • The entrywise conclusion applies in controller companion coordinates and is not invariant under general similarity transformations.Nonreal eigenvalues are allowed in conjugate pairs, but both A and Q must be real.Source: §2 · p. 2
  • Corollary 2.2 is stated under Theorem 2.1’s assumptions for the positive-definite condition Q ≻0; the supplied excerpt ends before giving its conclusion.Source: §2 · p. 2

Thus X is entrywise strictly positive as well as positive

The proof’s key adjugate-coefficient property is established by induction on matrix dimension.

Thus X is entrywise strictly positive as well as positive

3. Adjugate coefficients: a dimension-reduction proof

This section proves mixed positivity of adjugate coefficients by reducing a quadratic-form claim to a lower-dimensional accretive matrix and using induction.

3. Adjugate coefficients: a dimension-reduction proof

  • The dimension-reduction argument begins with a cofactor compression identity and an orthogonal basis adapted to a unit vector.The proof then applies Cauchy–Binet and cofactor identities to relate the compressed matrices.Source: §3 · pp. 2–3
  • The mixed result extends adjugate accretivity, recovered as the special case p = 0 and q = m − 1.That special case was proved by Zhang (2026, Proposition 4.3).Source: §3 · p. 3
  • The compression produces a real matrix B of one lower dimension satisfying B + B^T ⪰ 0.This accretivity condition is the basis for the subsequent coefficient argument.Source: §3 · p. 3
  • The proof uses a bivariate polynomial built from the Horner matrices and relates its coefficients to traces and adjugate terms.The determinant and adjugate identities used in the derivation remain valid even when the matrices are singular.Source: §3 · p. 3
  • Accretivity implies nonnegative coefficients for the characteristic polynomial of B, and induction then yields the mixed positivity claim for every unit vector.The eigenvalues lie in the closed right half-plane; real eigenvalues and nonreal conjugate pairs contribute factors with nonnegative coefficients.Source: §3 · p. 3

4. Proof of the continuous-time results

The proof uses the companion structure and a controllability Gramian to establish entrywise positivity of the continuous-time Lyapunov solution. Positive-semidefinite forcing gives nonnegative entries, while positive-definite forcing gives strictly positive entries.

4. Proof of the continuous-time results

  • The proof constructs Horner polynomial matrices that commute with A and uses uniqueness of the resulting Lyapunov equations.It then takes a Frobenius inner product with solutions of adjoint Lyapunov equations.Source: §4 · pp. 3–4
  • The companion structure yields a matrix C satisfying C + C^T ⪰ 0.Source: §4 · pp. 3–4
  • The Horner vectors span the state space, making (A, e_n) controllable and its Gramian positive definite.Hurwitz stability ensures convergence of the Gramian integral.Source: §4 · pp. 3–4
  • For positive-definite forcing, every entry of the Lyapunov solution is strictly positive.The proof establishes positivity for arbitrary k and ℓ.Source: §4 · p. 4

5. The discrete-time Lyapunov equation

The discrete-time Stein equation does not inherit the continuous-time entrywise nonnegativity result: Schur companion matrices can yield negative off-diagonal entries, even with identity forcing. The section gives a dimension-two counterexample and another with a polynomial whose coefficients are all positive.

5. The discrete-time Lyapunov equation

  • The section considers the direct discrete counterpart of the continuous-time equation and notes that the alternative convention replaces A with A^T.For Q ⪰ 0, the recalled Stein-equation result gives a unique symmetric solution, positive definite exactly under the stated observability condition.Source: §5 · p. 4
  • For the dimension-two example, (X2)12 = −16/25 < 0, and dimension two is minimal; in dimension one, |A| < 1 and Q ≥ 0 imply X = Q/(1−A2) ≥ 0.Source: §5 · pp. 4–5
  • Every strictly Schur pair with αβ < 0 gives a counterexample because the sign of X12 equals the sign of αβ.Source: §5 · p. 5
  • A strictly Schur companion matrix generated by a polynomial with strictly positive coefficients also has a negative solution entry under identity forcing.The example gives (X3)13 = −36/475 < 0; uniqueness and positive definiteness of the solution are also established.Source: §5 · p. 5
  • The Stein solution is positive semidefinite as a Gram matrix, but inner products between distinct functions may be negative.Source: §5 · p. 5

6. Positivity after a Cayley–companion transformation

A Cayley transform and state-basis change yield entrywise nonnegativity for the transformed Lyapunov solution of a Schur controller companion system. The basis is constructed from Horner polynomials and depends only on the state matrix; the conclusion does not extend automatically to the original coordinates.

6. Positivity after a Cayley–companion transformation

  • For a Schur controller companion matrix with Q ⪰ 0, the Cayley-transformed problem yields an entrywise nonnegative solution Y in a transformed companion basis.If Q ≻ 0, every entry of Y is strictly positive.Source: §6 · p. 5
  • Congruence by S turns the transformed forcing into a positive-semidefinite matrix, allowing the continuous-time companion theorem to establish Y’s nonnegativity.For Q ≻ 0, the transformed forcing is positive definite, yielding strict positivity of every entry of Y.Source: §6 · p. 6
  • The basis-change matrix S is built from Horner-polynomial columns, is nonsingular, and depends only on A, not on Q.The Horner recurrence gives HS = SB, with B the controller companion matrix of the characteristic polynomial.Source: §6 · p. 6
  • The positivity conclusion concerns Y in the new state basis; it does not imply entrywise nonnegativity of X in the original discrete companion coordinates.Entrywise nonnegativity is not preserved under general congruences.Source: §6 · p. 6

7. Conclusion

The paper proves entrywise positivity for continuous-time Lyapunov solutions of real Hurwitz controller companion matrices, with strict positivity under positive-definite forcing. It also establishes that the direct discrete-time analogue fails, while a Cayley transformation recovers positivity in a transformed companion basis.

7. Conclusion

  • For real Hurwitz controller companion matrices, Q ⪰ 0 gives an entrywise nonnegative Lyapunov solution, while Q ≻ 0 gives strict positivity.The proof uses mixed positivity of adjugate coefficients, dimension reduction, a controllability Gramian, and the Horner basis.Source: §7 · p. 6
  • The direct discrete-time analogue fails even with identity forcing and positive polynomial coefficients, but a Cayley transformation recovers nonnegativity in a state-matrix-dependent companion basis.The entrywise conclusions are specific to the controller companion coordinates.Source: §7 · p. 6
  • An open direction is to characterize which Schur companion matrices and positive-semidefinite forcings yield entrywise nonnegative Stein solutions in the original coordinates.Source: §7 · p. 6
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