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


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.


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.

Item Type:Thesis (Dissertation (Ph.D.))
Subject Keywords:Mathematics
Degree Grantor:California Institute of Technology
Division:Physics, Mathematics and Astronomy
Major Option:Mathematics
Thesis Availability:Public (worldwide access)
Research Advisor(s):
  • Taussky-Todd, Olga
Thesis Committee:
  • Unknown, Unknown
Defense Date:21 May 1976
Funding AgencyGrant Number
Record Number:CaltechTHESIS:04072017-144744158
Persistent URL:
Default Usage Policy:No commercial reproduction, distribution, display or performance rights in this work are provided.
ID Code:10133
Deposited By: Benjamin Perez
Deposited On:07 Apr 2017 22:41
Last Modified:05 Jan 2022 19:44

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