Strictly Anti-Diagonally Dominant Matrices in Systems of Yang–Baxter Matrix Equations

This study explores a system of Yang–Baxter-type matrix equations, XAX=BXB and XBX=AXA, which generalize the classical matrix Yang–Baxter equation. This work focuses on analyzing the existence of intertwining and commuting solutions using geometric and topological methods. To support this analysis, the notions of anti-diagonally dominant matrices (ADMs) and strictly anti-diagonally dominant matrices (SADMs) are introduced. It is shown that strictly anti-diagonally dominant matrices are nonsingular, ensuring stability and uniqueness in the associated linear systems. Furthermore, if the coefficient matrices of the system satisfy the SADM condition, then an intertwining solution X exists that fulfills both Yang–Baxter-type relations. When the matrices A and B are invertible, the corresponding solution X is proved to be a commuting one. These findings extend the algebraic framework of Yang–Baxter systems and provide new insights into the dominance properties that govern the solvability of matrix equations.

The equation was formally introduced by C. N. Yang in two landmark papers published in late 1967 on a one-dimensional quantum many-body problem. Yang established the form A(u)B(u+v)A(v) = B(u)A(u+v)B(v), where A(u) and B(v) are rational functions of the spectral parameters u and v. Later, in 1972, R. J. Baxter employed the same relation while solving the eight-vertex model in two-dimensional statistical mechanics.