Note on Hermitian structures of quadratic matrix expression via Yang-Baxter matrix equationS. GuerarraThe equation AXA=XAX, where X is unknown, is the original Yang-Baxter matrix equation for an arbitrary square matrix A. In this work, via the relation Q=AXA=XAX where A and Q are Hermitian matrices, we derived the extremal inertias of the quadratic matrix expression Q-XAX subject to the consistent matrix equation AXA=Q. As a consequence, we establish necessary and sufficient conditions for the solutions of the matrix equation AXA=Q to satisfy the quadratic matrix inequalitiesQ-XAX > (≥, <, ≤) 0 in the Löwner partial ordering, respectively. Advanced Studies: Euro-Tbilisi Mathematical Journal, Vol. 19(1) (2026), pp. 91-99 |