Weak majorization inequalities for the cubic and quartic coefficients of $e^{(A+B)t}$ versus $e^{At}e^{Bt}$

TL;DR

This paper establishes weak majorization inequalities comparing the eigenvalues of cubic and quartic coefficients of the exponential of a sum of matrices with the singular values of the corresponding coefficients of the product of matrix exponentials, using variational principles and commutator identities.

Contribution

It proves new weak majorization inequalities for the third and fourth order coefficients of matrix exponential sums versus products, extending known inequalities and providing conditions for higher orders.

Findings

Eigenvalues of $H^3$ and $H^4$ are weakly majorized by singular values of $Q_3$ and $Q_4$.

The proof combines Ky Fan variational principles with commutator identities.

A general sufficient condition for higher order inequalities is also provided.

Abstract

Let $A,B\in\mathbb{H}_n$ and set $H=A+B$. For each integer $k\ge 1$ define $$ Q_k:=\sum_{p=0}^k \binom{k}{p} A^pB^{k-p}, R_k:=\Re\,Q_k=\frac{Q_k+Q_k^*}{2}. $$ Then $H^k=\left.\frac{d^k}{dt^k}e^{Ht}\right|_{t=0}$ and $Q_k=\left.\frac{d^k}{dt^k}(e^{At}e^{Bt})\right|_{t=0}$. We prove that, for $k=3,4,$ $$ \lambda(H^k)\prec_w \sigma(Q_k). $$ Equivalently, the eigenvalues of the cubic and quartic Taylor coefficients of $e^{(A+B)t}$ are weakly majorized by the singular values of the corresponding coefficients of the Golden--Thompson product $e^{At}e^{Bt}$. Our argument combines Ky Fan variational principles with explicit commutator identitiesfor $R_k-H^k$ at orders $k=3,4$, reducing the problem to the positivity of certain double-commutator trace forms tested against Ky Fan maximizing projections. We also record a general sufficient condition for higher orders based on commutator decompositions.