The Matrix Equation F(A)X - XA = O

Citation

Parker, Joseph A. Jr. (1976) The Matrix Equation F(A)X - XA = O. Dissertation (Ph.D.), California Institute of Technology. doi:10.7907/X6W2-T505. https://resolver.caltech.edu/CaltechTHESIS:04072017-144744158

Abstract

In this work, all matrices are assumed to have complex entries. The cases of F(A) - XA = O where F(A) is a polynomial over C in A and F(A) = (A^*)^-1 are investigated. Canonical forms are derived for solutions X to these equations. Other results are given for matrices of the form A^-1A^*.

Let a set solutions {X_i} be called a tower if X_i+1 = F(X_i). 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, there is no polynomial for which only normal matrices A imply the existence of towers for all solutions X of P(A)X - XA = O. On the other hand, conditions are given for polynomials P, dependent upon spectrum of A, for which only diagonalizable matrices A imply the existence of towers for all solutions X of P(A)X - XA = O.

Thesis Files