On the matrix equation XHÂ =Â HX and the associated controllability problem

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 = HTX 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.