On a Procedure for Constructing A ⊗ A* Matrices of a Size 3 × 3 in Max-Plus Algebra

This paper presents a procedure for constructing a matrix A⊗A* of a size 3×3 in max-plus algebra. The max-algebraic product of a matrix A and its conjugate (negative transpose) matrix A* always yields a pseudo skew-symmetric matrix. We address the inverse problem: given a matrix A˜ of this type, can one determine a matrix A such that A⊗A*=A˜? A particular case involving 3×3 matrices is examined, and one procedure is explained in detail. The method illustrates how such a matrix A can be constructed column by column while also indicating directions for further generalization and computational exploration.