Abstract
Let <i>A</i> be a nonsingular matrix with only one Jordan block, we prove that the Yang-Baxter-like matrix equation <i>AXA</i> = <i>XAX</i> has no nonzero singular solution. When <i>A</i> is a nonsingular matrix with at least two Jordan blocks, the ranks of all nonzero singular solutions are obtained. This provides a necessary condition for a matrix to be a solution of the Yang-Baxter-like matrix equation. As applications, we obtain a family of nontrivial solutions for the nonsingular Jordan block with 3 × 3, and further investigate the non-commuting solutions for the nonsingular matrix with <i>n</i> × <i>n</i>.
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