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completely when A satisfies the equality A = I . The equation for the same class of the matrix A has been studied in our recent paper [9] under some additional assumption for the solutions. The contribution of the current paper is that we shall find all the solutions without any condition for them. The quadratic matrix equation (1) is called the YangBaxter matrix equation since it has been extensively studied several years ago from the viewpoint of linear algebra. The reason for the name of the equation is that it has a similarity in format to the famous Yang-Baxter equation in statistical mechanics [1, 10] initiated by Yang in 1967 and by Baxter in 1972. For applications of the Yang-Baxter equation to various areas of mathematics and physics, see, e.g., the monographs [7, 11] and the references therein. Since finding general solutions of the nonlinear matrix equation (1) is difficult, almost all the works so far have been toward constructing commuting solutions of the equation; see, e.g., [2, 3] and the references therein. In [5], all the commuting solutions were obtained when A is diagonalizable. In the more general case of A being nondiagonalizable, the structure of all commuting solutions has been obtained [6] for the class of nilpotent matrices. But still the explicit expression of all solutions of (1) is not available except for some special classes of matrices, such as that of idempotent matrices [8].

All non-commuting solutions of the Yang-Baxter matrix equation for a class of diagonalizable matrices