On the matrix equation XH=HX and the associated controllability problem

Abstract In this paper, a simple method for generating a nonsingular solution of the matrix equation XH  =  HX where the matrix H is in unreduced lower Hessenberg form is proposed. The controllability of the matrix pair ( H ,  b ) where b is the last column vector of solution matrix X is also addressed. In addition, a nonsingular symmetric solution of the matrix equation XH  =  H T X is obtained and the matrix pair ( H ,  b ) where b is the last row of the solution matrix of this equation is shown to be controllable.