THE GENERALIZED YANG-BAXTER MATRIX EQUATION A(XA)ᵐ=X(AX)ᵐ OVER M₂(F)

This paper investigates the generalized Yang-Baxter matrix equation A(XA)ᵐ=X(AX)ᵐ A ( X A ) m = X ( A X ) m for X∈ M₂(F) X ∈ M 2 ( F ) over a general field F F with char(F)≠ 2 char ( F ) ≠ 2 . Building upon our previous research [4] which characterized the set of nonsingular solutions SFᵐ(A)* S F m ( A ) ∗ for a nonsingular A , this study completes the classification by characterizing nonsingular solutions for singular A and describing the set of singular solutions SFᵐ(A)∘ S F m ( A ) ∘ for arbitrary 2× 2 2 × 2 matrices A . Consequently, these results provide a comprehensive description of the full solution set SFᵐ(A) S F m ( A ) for all 2× 2 2 × 2 matrices and positive integers m .