A Characterization for Spectral and Trace Positivity of Matrix Words

Abstract

In this paper, we provide a complete structural characterization of which words in noncommuting matrix variables and their formal transposes universally evaluate to matrices with nonnegative eigenvalues or nonnegative trace, regardless of the dimension or entries of the substituted matrices. Crucially, we establish that in this pure monomial setting, both spectral and trace positivity exhibit an exact rigidity phenomenon analogous to Hermitian-square representations. This contrasts sharply with the broader polynomial setting, where tracial rigidity is known to fail even for approximation. Furthermore, we establish that this structural rigidity extends seamlessly to the complex and infinite-dimensional domains. We show that the exact same purely combinatorial conditions completely characterize matrix words evaluated over complex matrices, bounded operators on arbitrary Hilbert spaces, and positive tracial states on $C^*$-algebras and von Neumann algebras. As one application of our main theorems, we fully resolve open questions initially posed by Lieb and Pierce, and later conjectured by Hillar and Johnson, concerning the underlying algebraic structure of the summands in Bessis-Moussa-Villani (BMV) trace polynomials from quantum statistical mechanics. We achieve these results by establishing a surprising connection to graph theory, leading to a matrix analogue of the positive graph conjecture posed by Camarena, Csóka, Hubai, Lippner, and Lovász.