Diagonalization of Matrices
DIAGONALIZATION OF MATRICES AMAJOR QUALIFYING PROJECT REPORT SUBMITTED TO THE FACULTY OF WORCESTER POLYTECHNIC INSTITUTE IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF BACHELOR OF SCIENCE WRITTEN BY THEOPHILUS MARKS APPROVED BY:PADRAIG OC´ ATHAIN´ MAY 5, 2021 THIS REPORT REPRESENTS THE WORK OF WPI UNDERGRADUATE STUDENTS SUBMITTED TO THE FACULTY AS EVIDENCE OF COMPLETION OF A DEGREE REQUIREMENT. 1 Abstract This thesis aims to study criteria for diagonalizability over finite fields. First we review basic and major results in linear algebra, including the Jordan Canonical Form, the Cayley-Hamilton theorem, and the spectral theorem; the last of which is an important criterion for for matrix diagonalizability over fields of characteristic 0, but fails in finite fields. We then calculate the number of diagonalizable and non-diagonalizable 2 × 2 matrices and 2 × 2 symmetric matrices with repeated and distinct eigenvalues over finite fields. Finally, we look at results by Brouwer, Gow, and Sheekey for enumerating symmetric nilpotent matrices over finite fields. 2 Summary (1) In Chapter 1 we review basic linear algebra including eigenvectors and eigenval- ues, diagonalizability, generalized eigenvectors and eigenspaces, and the Cayley- Hamilton theorem (we provide an alternative proof in the appendix). We explore the Jordan canonical form and the rational canonical form, and prove some necessary and sufficient conditions for diagonalizability related to the minimal polynomial and the JCF. (2) In Chapter 2 we review inner products and inner product spaces, adjoints, and isome- tries. We prove the spectral theorem, which tells us that symmetric, orthogonal, and self-adjoint matrices are all diagonalizable over fields of characteristic zero.
[Show full text]