<i>J</i> -symmetric solutions of <i>AX</i>  =  <i>B</i> over Bézout domains: a simultaneous basis approach

Let R be a commutative Bézout domain. We investigate (J,ε)-symmetric solutions of the linear matrix equation AX = B, characterized by XTJ=εJX for a fixed symmetric unimodular bilinear form J∈GL⁡(n,R) and ε∈{+1,−1}. We establish a sharp existence criterion for such constrained solutions, recovering Prokip's classical symmetric solvability theorem and its skew-symmetric analogue as natural special cases. Our primary contribution is a simultaneous (A,J)-adapted basis theorem: under a natural nondegeneracy hypothesis on Ker⁡(A), we explicitly construct unimodular transformation matrices that simultaneously reduce A to a nonsingular diagonal block form and block-diagonalize J. This construction is driven by a novel J-Bézout Gram–Schmidt orthogonalization algorithm, yielding a transparent block parametrization of the entire solution space. Finally, we establish that determinantal ideals of tI−X and localized primary isometry data form robust invariants under O(J,R)-similarity, and prove a conditional reconstruction theorem for strongly split semisimple operators over arbitrary Bézout domains.