МАТРИЧНОЕ УРАВНЕНИЕ AX+XB=C НАД КОНЕЧНЫМ ПОЛЕМ

Abstract

The matrix equation AX+XB=C over a finite Galois field is considered, and the questions of the solvability of this equation are investigated. To analyze the solvability, the equation is transformed to the form A ̃X ̃+ X ̃B ̃=C ̃, where the matrices A ̃ and B ̃ have Jordan normal form over the decomposition field of the product of the characteristic polynomials of the matrices A and B. This allows us to consider the last equation in the form of a system of block equalities. For this, the matrices A ̃, B ̃, C ̃, X ̃ are partitioned into blocks A ̃αβ , B ̃αβ , C ̃αβ , X ̃αβ , each of which corresponds to only one Jordan box with an eigenvalue λα of the matrix A and only one Jordan box with an eigenvalue μβ of the matrix B. To find the unknown blocks, the Frobenius iteration method is used. If the characteristic polynomials of the matrices A and -B are coprime, the solution to the matrix equation exists and is unique. If λα+μβ=0 is true for some pairs of roots from the decomposition field of the product the characteristic polynomials of the matrices A and B, then the solution exists if and only if the solvability condition is satisfied. Examples are given to illustrate the obtained results.