# Determinants and Eigenvalues

Massive Open Online Course
• Overview
• Course Content
• Requirements & Materials
Overview

## Determinants and Eigenvalues

Course Description

At the beginning of this course we introduce the determinant, which yields two important concepts that you will use in this course. First, you will be able to apply an invertibility criterion for a square matrix that plays a pivotal role in, for example, the understanding of eigenvalues. You will also use the determinant to measure the amount by which a linear transformation changes the area of a region. This idea plays a critical role in computer graphics and in other more advanced courses, such as multivariable calculus.

This course then moves on to eigenvalues and eigenvectors. The goal of this part of the course is to decompose the action of a linear transformation that may be visualized. The main applications described here are to discrete dynamical systems, including Markov chains. However, the basic concepts— eigenvectors and eigenvalues—are useful throughout industry, science, engineering, and mathematics.

Prospective learners enrolling in this course are encouraged to first complete the linear equations and matrix algebra courses before starting this class.

Course Content

INTRODUCTION TO DETERMINANTS

PROPERTIES OF THE DETERMINANT

VOLUME AND LINEAR TRANSFORMATIONS

MARKOV CHAINS

EIGENVALUES AND EIGENVECTORS

THE CHARACTERISTIC EQUATION

DIAGONALIZATION

COMPLEX EIGENVALUES

Requirements & Materials
Prerequisites

Recommended

• High school algebra, geometry, and pre-calculus

Required

• Linear Equations (DL 0050M)

• Matrix Algebra (DL 0051M)

Materials

Required

• Internet connection (DSL, LAN, or cable connection desirable)

### Who Should Attend

This course is designed for undergraduate students, advanced high school students, who are interested in pursuing any career path or degree program that involves linear algebra, or industry employees who are seeking a better understanding of linear algebra for their career development. ### What You Will Learn

• How to compute determinants of using cofactor expansions and properties of determinants
• Compute areas and volumes of regions in Rn under a given linear transformation using determinants
• Real-world problems using Markov chains
• The characteristic polynomial to compute eigenvalues of a matrix
• Eigenvalues and the corresponding eigenspaces for a matrix
• The invertibility of a matrix using determinants and eigenvalues
• How to decompose matrices into a product of matrices by using a matrix factorization
• Eigenvalues to determine identify the rotation and dilation of a linear transform
• How to construct the Google matrix for web and compute its Google PageRank ### How You Will Benefit

• Apply theorems related to eigenvalues to characterize the invertibility of a matrix.
• Apply theorems to characterize matrices with complex eigenvalues.
• Apply matrix powers and theorems to characterize the long-term behavior of a Markov chain.
• Construct a transition matrix, a Markov Chain, and a Google Matrix for a given web, and compute the PageRank of the web.
• Analyze mathematical statements and expressions involving linear systems and matrices. For example, to describe matrices in terms of diagonalizability and their eigenvectors.
• ##### Taught by Experts in the Field
•  - Abe Kani
President