<p>Let the Jordan canonical form of a nonsingular <InlineEquation ID="IEq3"> <EquationSource Format="TEX">$3 \times 3$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mn>3</mn> <mo>×</mo> <mn>3</mn> </mrow> </math></EquationSource> </InlineEquation> matrix <i>A</i> be <InlineEquation ID="IEq4"> <EquationSource Format="TEX">$J=J_3[\lambda ]$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>J</mi> <mo>=</mo> <msub> <mi>J</mi> <mn>3</mn> </msub> <mrow> <mo stretchy="false">[</mo> <mi>λ</mi> <mo stretchy="false">]</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation>, <InlineEquation ID="IEq5"> <EquationSource Format="TEX">$\lambda \ne 0$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>λ</mi> <mo>≠</mo> <mn>0</mn> </mrow> </math></EquationSource> </InlineEquation>. We find all non-trivial solutions of the Yang–Baxter-like matrix equation <InlineEquation ID="IEq6"> <EquationSource Format="TEX">$JYJ=YJY$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>J</mi> <mi>Y</mi> <mi>J</mi> <mo>=</mo> <mi>Y</mi> <mi>J</mi> <mi>Y</mi> </mrow> </math></EquationSource> </InlineEquation> in this paper. Based on this result, we can use the similarity transformation and obtain all non-trivial solutions of Yang–Baxter-like matrix equation <InlineEquation ID="IEq7"> <EquationSource Format="TEX">$AXA=XAX$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>A</mi> <mi>X</mi> <mi>A</mi> <mo>=</mo> <mi>X</mi> <mi>A</mi> <mi>X</mi> </mrow> </math></EquationSource> </InlineEquation>. We also investigate the intrinsic structure of the non-trivial solution set and prove that it is path-connected. Additionally, using our approach, we can find solutions to the Yang–Baxter-like equation with a single parameter.</p>

All solutions of the Yang–Baxter-like matrix equation for a nonsingular $3\times 3$ Jordan block

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发表日期：

2025年10月15日

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Xiaoli Li,
Lanping Zhu,
Qianglian Huang

摘要

Let the Jordan canonical form of a nonsingular $3 \times 3$ $3 \times 3$ matrix A be $J=J_3[\lambda ]$ $J = J_{3} [λ]$ , $\lambda \ne 0$ $λ \neq 0$ . We find all non-trivial solutions of the Yang–Baxter-like matrix equation $JYJ=YJY$ $J Y J = Y J Y$ in this paper. Based on this result, we can use the similarity transformation and obtain all non-trivial solutions of Yang–Baxter-like matrix equation $AXA=XAX$ $A X A = X A X$ . We also investigate the intrinsic structure of the non-trivial solution set and prove that it is path-connected. Additionally, using our approach, we can find solutions to the Yang–Baxter-like equation with a single parameter.

All solutions of the Yang–Baxter-like matrix equation for a nonsingular \(3\times 3\) Jordan block

摘要