On the BMV conjecture for 2\times2 matrices and the exponential convexity of the function \cosh(\sqrt{at^2+b})

TL;DR

This paper provides a simple matrix-based proof of the BMV conjecture for 2x2 matrices, demonstrating exponential convexity of a specific hyperbolic cosine function using Lie formulas and Pauli matrices.

Contribution

It offers a novel, purely matrix-analytic proof of the BMV conjecture for 2x2 matrices, avoiding complex algebraic geometry methods.

Findings

Proof confirms exponential convexity for 2x2 case

Utilizes Lie product formula and Pauli matrices

Simplifies understanding of the BMV conjecture in low dimensions

Abstract

The BMV conjecture states that for \(n\times n\) Hermitian matrices \(A\) and \(B\) the function \(f_{A,B}(t)=\tr e^{tA+B}\) is exponentially convex. Recently the BMV conjecture was proved by Herbert Stahl. The proof of Herbert Stahl is based on ingenious considerations related to Riemann surfaces of algebraic functions. In the present paper we give a purely "matrix" proof of the BMV conjecture for \(2\times2\) matrices. This proof is based on the Lie product formula for the exponential of the sum of two matrices. The proof also uses the commutation relations for the Pauli matrices and does not use anything else.