[Paper Review] Advances on the Bessis-Moussa-Villani Trace Conjecture

Christopher J. Hillar|ArXiv.org|Jul 8, 2005

Matrix Theory and Algorithms9 references4 citations

TL;DR

This paper advances the Bessis-Moussa-Villani (BMV) trace conjecture by deriving Euler-Lagrange equations for matrices minimizing or maximizing a coefficient of the trace polynomial Tr[(A + tB)^m]. It proves that if the conjecture fails for some m, it fails for all larger m, and shows that verifying the conjecture for singular positive semidefinite matrices or infinitely many m suffices to prove it generally.

ABSTRACT

A long-standing conjecture asserts that the polynomial \[p(t) = ext{Tr}[(A+tB)^m]\] has nonnegative coefficients whenever $m$ is a positive integer and $A$ and $B$ are any two $n imes n$ positive semidefinite Hermitian matrices. The conjecture arises from a question raised by Bessis, Moussa, and Villani (1975) in connection with a problem in theoretical physics. Their conjecture, as shown recently by Lieb and Seiringer, is equivalent to the trace positivity statement above. In this paper, we derive a fundamental set of equations satisfied by $A$ and $B$ that minimize or maximize a coefficient of $p(t)$. Applied to the Bessis-Moussa-Villani (BMV) conjecture, these equations provide several reductions. In particular, we prove that it is enough to show that (1) it is true for infinitely many $m$, (2) a nonzero (matrix) coefficient of $(A+tB)^m$ always has at least one positive eigenvalue, or (3) the result holds for singular positive semidefinite matrices. Moreover, we prove that if the conjecture is false for some $m$, then it is false for all larger $m$.

Motivation & Objective

• To resolve the long-standing Bessis-Moussa-Villani (BMV) trace conjecture, which asserts that Tr[(A + tB)^m] has nonnegative coefficients for positive semidefinite Hermitian matrices A and B.
• To identify necessary conditions for extremal matrices A and B that minimize or maximize a coefficient of the trace polynomial p(t) = Tr[(A + tB)^m].
• To reduce the conjecture to simpler cases by proving that if it fails for some m, it fails for all larger m, and that it suffices to verify it for singular matrices or infinitely many m.
• To establish that the conjecture's validity for singular positive semidefinite matrices implies its validity for all such matrices, including invertible ones.

Proposed method

• Derives a fundamental pair of matrix equations—referred to as Euler-Lagrange equations—by applying variational calculus to the trace coefficient Tr[S_{m,k}(A,B)] under unit norm constraints.
• Uses the trace identity Tr[S_{m,k}(A,B)] = (m/(m-k)) Tr[AS_{m-1,k}(A,B)] = (m/k) Tr[BS_{m-1,k-1}(A,B)] to relate coefficients across different m and k.
• Applies the Cauchy-Schwarz inequality and norm bounds to derive the upper bound |Tr[S_{m,k}(A,B)]| ≤ (m choose k) for Hermitian matrices of unit norm.
• Employs the fact that the product of positive semidefinite matrices has nonnegative eigenvalues to analyze the sign of Tr[AS_{m,k}(A,B)] and Tr[BS_{m,k}(A,B)].
• Uses homogeneity and continuity arguments to reduce the problem to matrices of unit norm and to analyze extremal configurations.
• Applies contradiction arguments based on trace identities and eigenvalue nonnegativity to prove that invertible matrices cannot minimize the trace coefficient unless the conjecture holds.

Experimental results

Research questions

• RQ1 Under what conditions do matrices A and B minimize or maximize a coefficient of the trace polynomial Tr[(A + tB)^m]?
• RQ2 If the BMV conjecture fails for some m, does it fail for all larger m?
• RQ3 Can the conjecture be reduced to the case of singular positive semidefinite matrices?
• RQ4 Is it sufficient to verify the conjecture for infinitely many m to establish its general validity?
• RQ5 What constraints do the Euler-Lagrange equations impose on extremal matrices A and B?

Key findings

• If the BMV conjecture fails for some m, then it fails for all larger m, establishing an upward closure property of counterexamples.
• The conjecture holds for all m if and only if it holds for infinitely many m, providing a strong reduction criterion.
• The conjecture is equivalent to the statement that every nonzero coefficient of (A + tB)^m has at least one positive eigenvalue when A and B are positive semidefinite.
• The conjecture holds for all positive semidefinite matrices if and only if it holds for all singular positive semidefinite matrices.
• If a coefficient Tr[S_{m,k}(A,B)] is minimized at a point where A and B are invertible, a contradiction arises, implying that extremal matrices must be singular.
• The upper bound |Tr[S_{m,k}(A,B)]| ≤ (m choose k) holds for all Hermitian matrices of unit norm, with equality only when A = ±B and A has a single nonzero eigenvalue for m > 2.