Matrix Methods Syllabus

COURSE INFORMATION

Course Description: Matrices and matrix algebra, determinants, systems of linear equations,
Gaussian elimination , eigenvalues and eigenvectors, linear transformation , applications in science
and engineering.

Course Prerequisites : Math 2414 or concurrent enrollment

Student Learning Outcomes: Upon completion of this course, students should be able to do the
following:
• Perform basic matrix operations including row reduction , transpose, finding the inverse
and finding the determinant.
• Solve systems of linear equations using substitution, Gaussian elimination, Cramers rule
and inverse matrices.
• Find eigenvalues and eigenvectors as well as understanding their properties and importance
to matrix theory and applications.
• Understand the basic properties of Euclidean space including linear independence, dimension,
rank, orthoganality, norm and projection.

COURSE CONTENT

Required Text:
Bronson, Richard. Matrix Methods , An Introduction , 2nd edition, Academic Press, 1991.
ISBN: 978-0121352516

List of Course Topics: The following is a detailed list of the sections from your textbook. Chapters
one through five will be covered in detail and selected topics will be chosen from the remaining
chapters.

Chapter 1. Matrices
Section 1.1: Basic Concepts
Section 1.2: Operations
Section 1.3: Matrix Multiplication
Section 1.4: Special Matrices
Section 1.5: Submatrices and Partitioning
Section 1.6: Vectors
Section 1.7: The Geometry of Vectors

Chapter 2. Simultaneous Linear Equations
Section 2.1: Linear Systems
Section 2.2: Solutions by Substitution
Section 2.3: Gaussian Elimination
Section 2.4: Pivoting Strategies
Section 2.5: Linear Independence
Section 2.6: Rank
Section 2.7: Theory of Solutions

Chapter 3. The Inverse
Section 3.1: Introduction
Section 3.2: Calculating Inverses
Section 3.3: Simultaneous Equations
Section 3.4: Properties of the Inverse
Section 3.5: LU Decomposition

Chapter 4. Determinants
Section 4.1: Introduction
Section 4.2: Expansion by Cofactors
Section 4.3: Properties of Determinants
Section 4.4: Pivotal Condensation
Section 4.5: Inversion
Section 4.6: Cramer’ s Rule

Chapter 5. Eigenvalues and Eigenvectors
Section 5.1: Definitions
Section 5.2: Eigenvalues
Section 5.3: Eigenvectors
Section 5.4: Properties of Eigenvalues and Eigenvectors
Section 5.5: Linearly Independent Eigenvectors
Section 5.6: Power Methods

Chapter 6. Real Inner Products
Section 6.1: Introduction
Section 6.2: Orthonormal Vectors
Section 6.3: Projections and QR-Decomposition
Section 6.4: The QR-Algorithm
Section 6.5: Least- Squares

Chapter 7. Matrix Calculus
Section 7.1: Well-Defined Functions
Section 7.2: Cayley-Hamilton Theorems
Section 7.3: Polynomials of Matrices – Distinct Eigenvalues
Section 7.4: Polynomials of Matrices – General Case
Section 7.5: Functions of a Matrix
Section 7.6: The Function eAt
Section 7.7: Complex Eigenvalues
Section 7.8: Properties of eA
Section 7.9: Derivatives of a Matrix

COURSE TIMELINE

Tentative Schedule: This schedule should give you a rough idea of the pace of the course. Due to
the accelerated nature of summer courses , you will be expected to cover a great deal of material
in each class, and content will build quickly. It is extremely important to stay current with course
topics.

COURSE EVALUATION

Homework: You will have assigned homework every night. Due to the accelerated timeline of
the course and the amount of material to be covered, you MUST stay on top of your homework in
order to succeed in this course. Homework will be collected two class days after assigned unless
otherwise announced in class. For submission, fold homework lengthwise (into a 4” ×11”) and place
on the teaching podium BEFORE CLASS BEGINS. Late homework will not be accepted.

In-Class Exams: You will have three in-class exams, tentatively scheduled for 22 July, 31 July,
and 12 August. As per the technology policy, no technological assistance will be available during
exams.

Final Exam: You will have a comprehensive final exam during the regularly scheduled class on
Friday, 14 August.

Grading: Grades will be computed as follows:

IMPORTANT DATES

INSTRUCTOR’S POLICIES

Make-ups: Make-ups for documented absences that are required as part of a UT Tyler obligation
(e.g., athletes participating in an event, participation in a debate contest, etc.) or for religious observation
will be granted. For all make-ups of this type, documentation and prior notification of
at least one week are required. All other make-ups will be granted only in extreme circumstances
and solely at Dr. Graves’s discretion.

Technology Policy: No calculators, PDAs, or computers will be allowed on exams. Use of Mathematica
or other software may be helpful in checking homework answers, but credit will only be
given for homework showing legible handwritten work.

UNIVERSITY POLICIES

For University policies concerning Students’ Rights and Responsibilities, Grade Replacement/Forgiveness,
State-Mandated Course Drop Policy, Disability Services, Student Absence due to Religious Observance,
Student Absence for University-Sponsored Events and Activities, and the Social Security
and FERPA Statement please see: