Solving the Yang–Baxter-like matrix equation for nilpotent matrices of index three

ABSTRACT

Let A be a nilpotent matrix of index three and consider the Yang–Baxter-like matrix equation AXA=XAX. We first obtain a simplified matrix equation of the same type with A replaced by its Jordan form, whose Jordan blocks are at most $3\times 3$. Then we obtain a system of matrix equations of smaller sizes to find all the commuting solutions of the original matrix equation.

Keywords:

• Nilpotent matrix
• index
• matrix equation
• Jordan form
• commutability

AMS Classifications:

• 15A24
• 65F15

Disclosure statement

No potential conflict of interest was reported by the authors.

Additional information

Funding

This work was supported by the National Natural Science Foundation of China (Nos. 11501126, 11471122), the Youth Natural Science Foundation of Jiangxi Province (No. 20151BAB211011), the Science Foundation of Jiangxi Provincial Department of Education (No. GJJ150979), the Local Colleges and Universities Development Supporting Foundation of China – Applied Mathematics Innovative Team Building, and the 111 Project.