The matrix equation \(XA-BX=R\) and its applications
The author considers the Sylvester equation \(XA-BX=R\) for X where A, B and R are \(n\times n\) complex matrices, and B is nonderogatory. The main theorem shows that if only the first n-1 rows of R are given, then there is at most one solution X with a specified first row, and the author characterizes the case where X is nonsingular. This result is used to construct symmetrizers and commuting matrices, to compute the number of common eigenvalues between A and B, and to reprove some results on symmetrizers, Bézoutians and inertia.
scientific article; zbMATH DE number 4071107