Solving Yang-Baxter Matrix Equation via Extremal Ranks of Partial Banded Block Matrix

The equation $ABA=BAB$, where $B$ is unknown matrix, is the original Yang-Baxter matrix equation for an arbitrary square matrix $A$. In this work, we establish the formulas of the extremal ranks of a $3\times3$ partial banded block matrix% \[ \left[ \begin{array} [c]{ccc}% M_{11} & M_{12} & X\\ M_{12}^{\ast} & M_{22} & M_{23}\\ X & M_{23}^{\ast} & M_{33}% \end{array} \right] \] where $X$ is a variant complex matrix subject to the linear matrix equation $AXA=T$, where $A$ and $T$ are Hermitian matrices. To clarify the practical aspects of the reached results, we provide a necessary and sufficient condition for the solvability of the Yang Baxter matrix equation $AXA=XAX=T$ over the complex field $% %TCIMACRO{\U{2102} }% %BeginExpansion \mathbb{C} %EndExpansion .$.