The matrix equation XA − BX = R and its applications

Abstract We study the well-known Sylvester equation XA − BX = R in the case when A and B are given and R is known up to its first n - 1 rows. We prove new results on the existence and uniqueness of X . Our results essentially state that, in case A is a nonderogatory matrix, there always exists a solution to this equation; a solution is uniquely determined by its first row x 1 ; and there is an interesting relationship between x 1 and the rows of R . We also give a complete characterization of the nonsingularity of X in this case. As applications of our results we develop direct methods for constructing symmetrizers and commuting matrices, computing the characteristic polynomial of a matrix, and finding the numbers of common eigenvalues between A and B . Some well-known important results on symmetrizers, Bezoutians, and inertia are recovered as special cases.