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

Author: Parker, Joseph A. Jr.

Year: 1976

Degree: Dissertation (Ph.D.)

Advisor: Taussky-Todd, Olga

Committee Member: Unknown, Unknown

Option: Mathematics

DOI: 10.7907/X6W2-T505

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 {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, 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.

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