On symmetric solutions of the matrix equation AX = B over a Bezout domain

Abstract

Abstract. Let Rm,n be the set of m × n matrices over a Bezout domain R with identity e ̸= 0, and let 0m,k be the zero m×k matrix. Further, let di(A) ∈ R be an ideal generated by the i−th order minors of the matrix A ∈ Rm,n, i = 1, 2, . . . , min{m, n}. The rank of a matrix A, denoted by rank A, is the highest order of a non-zero minor of the matrix A. (The rank of the zero matrix is 0.) The transpose matrix of a matrix A ∈ Rm,n will be denoted by AT . The square matrix A is symmetric if A = AT . In what follows C ∗ = Adj (C) means the classical adjoint matrix for a nonsingular matrix C ∈ Rn,n, i.e. C ∗C = In det C. In this report we investigate a structure of solutions of a matrix equation AX = B, (1) where A ∈ Rm,n and B ∈ Rm,k are known matrices and X is unknown matrix over R. Put AB = [A, B] ∈ Rm,n+k. It is known (see [4]) that equation (1) is solvable over a Bezout domain if and only if rank A = rank AB = r and di(A) = di(AB) for all i = 1, 2, . . . , r. On the other hand AX = B is solvable over R if and only if matrices [A, 0m,k] and AB are right-equivalent, that is, the Hermitian normal forms of these matrices coincide [5].