#### Date of Award

8-1966

#### Document Type

Thesis

#### Degree Name

Master of Science

#### Department

Master of Mathematics

#### Abstract

The term matrix was first used in 1850 by J. J. Sylvester. 1858 Arthur Cay ley began the systematic development of the theory of Today, matrix theory has application in such diverse fields as inventory control in factories, quantum theory in physics cost In matrices. analysis in transportations and other industries, deployment problems in military operations, and data analysis in sociology and psychology.

This paper concerns itself with the set of necessary and sufficient conditions which will insure certain or particular canonical forms of matrices. Canonical forms of matrices are an important phenomena in the study of Differential Equations and other mathematical courses.

In this paper, the development of the canonical form of a matrix under similarity transformation shall be presented. Letting A be a n-square matrix with elements in the field F, where F is a real number field, and let P be a non-singular matrix with elements in F. The set of all matrices P -1 AP constitutes a class of matrices similar to A. These matrices are known as canonical forms. In other words, a matrix of order nxn will be transformed to a canonical form of a matrix. If A and B denote two n-square matrices with elements in The necessary matrix. F such that P_1AP = B, A and B are said to be similar, and sufficient conditions that A and B be similar are that the

X-matrices A- Xl and B- XI have the same invariant factor and the same elementary divisors.

Chapter I includes Notations and Definitions. Chapter II includes a set of Basic Theorems of matrices to be used in the development of the canonical forms.

In Chapter III five canonical forms of matrices of order nxn shall be developed. These forms are Diagonal, Triangular, Rational, Jordon, and Smith Normal Form.

In Chapter IV some examples of the canonical forms in Chapter III will be presented

#### Committee Chair/Advisor

A. D. Stewart

#### Publisher

Prairie View Agricultural and Mechanical College

#### Rights

© 2021 Prairie View A & M University

This work is licensed under a Creative Commons Attribution-NonCommercial 4.0 International License.

#### Date of Digitization

12/01/2021

#### Contributing Institution

John B Coleman Library

#### City of Publication

Prairie View

#### MIME Type

Application/PDF

#### Recommended Citation

Steward, J. S. (1966). On A Set Of Classical Canonical Forms For An NxN Matrix. Retrieved from https://digitalcommons.pvamu.edu/pvamu-theses/805