All Projection-Based Commuting Solutions of the Yang-Baxter-like Matrix Equation
Abstract
We determine all the commuting solutions of the quadratic matrix equation $AXA=XAX$, which
can be expressed in terms of projection matrices. All such solutions have special structures
and their building blocks satisfy a system of matrix equations, and our general result extends
the previous ones in {\it J. Math. Anal. Appl.} 402 (2013), 567-573, {\it Applied
Math. Lett.} 35 (2014), 86-89, {\it Applied Math. Lett.} 64 (2017), 231-234, and
{\it Applied Math. Lett.} 79 (2018), 155-161.
can be expressed in terms of projection matrices. All such solutions have special structures
and their building blocks satisfy a system of matrix equations, and our general result extends
the previous ones in {\it J. Math. Anal. Appl.} 402 (2013), 567-573, {\it Applied
Math. Lett.} 35 (2014), 86-89, {\it Applied Math. Lett.} 64 (2017), 231-234, and
{\it Applied Math. Lett.} 79 (2018), 155-161.
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