In this work, all matrices are assumed to have complex entries. The cases ofF(A) - XA = O where F(A) is a polynomial over C in A andF(A) = (A*)-1 are investigated. Canonical forms are derived forsolutions X to these equations. Other results are given for matricesof the form A-1A*.
Let a set solutions {Xi} be called a tower if Xi+1 = F(Xi).It is shown that towers occur for all nonsingular solutions of(A*)-1X - XA = O if and only if A is normal. In contrast to this, thereis no polynomial for which only normal matrices A imply the existence oftowers for all solutions X of P(A)X - XA = O. On the other hand, conditionsare given for polynomials P, dependent upon spectrum of A, forwhich only diagonalizable matrices A imply the existence of towers for all solutionsX of P(A)X - XA = O.
