# Matrix Algebra

This course will teach you the basics of vector and matrix algebra and operations necessary to understand multivariate statistical methods, including the notions of the matrix inverse, generalized inverse and eigenvalues and eigenvectors.

## Overview

Statistics deals with collections of data organized in 1,2,3 or more dimensions. Matrix notation is the best way to compactly represent such data. This course provides the basics of vector and matrix algebra and operations necessary to understand multivariate statistical methods including the matrix inverse, generalized inverse and eigenvalues and eigenvectors.

• Intermediate
• 4 Weeks
• Expert Instructor
• Tuiton-Back Guarantee
• 100% Online
• TA Support

## Learning Outcomes

Students who complete this course will be introduced to vectors and matrices, matrix inverse and linear equations, Eigenvalues and Eigenvectors, and symmetric matrices. It is designed to be a refresher, or an introduction to matrix methods for anyone interested in multivariate statistics, modeling, designing experiments or data mining.

• Define scalars, vectors, matrices and arrays, and use proper notation
• Conduct vector and matrix operations
• Calculate order and rank for a matrix
• Find the eigenvalues and eigenvectors of a real, square matrix
• Solve a system of linear equations and the generalized inverse

## Who Should Take This Course

Matrix algebra is used heavily in multivariate statistics, and the theory behind many statistical modeling procedures. Matrix notation is used even more widely. If you are interested in taking courses in multivariate statistics, modeling, design of experiments, data mining or other topics involving multivariate data and need a refresher in, or introduction to matrix methods, you should take this course.

## Our Instructors

#### Dr. Robert LaBudde

Dr. Robert LaBudde is president and founder of Least Cost Formulations, Ltd., a mathematical software development company specializing in optimization and process control software for manufacturing companies. He has served on the faculties of the University of Wisconsin, Massachusetts Institute of Technology, Old Dominion University and North Carolina State University. Dr. LaBudde is currently Adjunct Professor of Statistics at Old Dominion University.

## Course Syllabus

### Week 1

Introduction to Vectors and Matrices

• Notation
• Definitions of scalars, vectors, matrices and arrays
• Vector and matrix operations and the transpose
• Inner and outer products
• Zero and Identity matricesMatrix multiplication
• Order and rank of a matrix
• Length, norm and distance
• Angle between two vectors, orthogonality

### Week 2

Matrix Inverse & linear Equations

• Order and rank of a matrix
• Elementary row and column operations
• Row and column echelon forms
• Inverse of a square matrix
• Applications to statistics
• Linear combinations, dependence and independence
• More on the rank of a matrix
• The generalized inverse
• Homogeneous equations
• Solving a system of linear equations and the generalized inverse
• Determinant of a square matrix
• Applications of determinants in statistics

### Week 3

Eigenvalues and Eigenvectors

• The characteristic equation and eigenvalues and eigenvectors of a real, square matrix
• Finding eigenvalues and eigenvectors of a matrix
• Geometric interpretation

### Week 4

Symmetric matrices

• Symmetric matrices
• Positive definite, semi-definite and non-negative definite matrices
• Eigenvalues and eigenvectors of a real symmetric matrix
• The spectral decomposition of a symmetric matrix
• Principal components analysis
• Applications to statistics

## Class Dates

### 2023

08/11/2023 to 09/08/2023
Instructors: Dr. Robert LaBudde

### 2024

03/22/2024 to 04/19/2024
Instructors: Dr. Robert LaBudde
08/09/2024 to 09/06/2024
Instructors: Dr. Robert LaBudde

### 2025

03/21/2025 to 04/11/2025
Instructors: Dr. Robert LaBudde
08/08/2025 to 09/05/2025
Instructors: Dr. Robert LaBudde

## Prerequisites

You should be familiar with intermediate or college algebra, including solving systems of linear equations.

The additional preparation found in Introduction to Statistics 3 – Regression and ANOVA is also helpful.

## Register For This Course

Matrix Algebra

#### Homework

Homework in this course consists of guided numerical problems to test the concepts.

In addition to assigned readings, this course also has the instructor’s expert write-ups on important concepts.

#### Course Text

All course materials will be provided in the course.

For those who wish additional resources, a recommended text is Matrix Algebra: An Introduction by Krishnan Namboodiri from Sage. Sage Publication offers discounts to students at Statistics.com for many of their titles when the code S06SC is used during checkout on their website (the “0” is a zero not an alphabetical “O”). If you are located in Asia, the web procedure for your location may not accept this discount, try calling your regional Sage representative.

#### Software

There is no requirement for software in this course. All of the assignments can be done by hand. However, the text illustrates some examples using SAS and the course notes using Microsoft Excel and R.

#### Course Fee & Information

Enrollment
Courses may fill up at any time and registrations are processed in the order in which they are received. Your registration will be confirmed for the first available course date unless you specify otherwise.

Transfers and Withdrawals
We have flexible policies to transfer to another course or withdraw if necessary.

Group Rates

Discounts
Academic affiliation?  In most courses you are eligible for a discount at checkout. Use promo code ACADEMIC where prompted during registration.

Invoice or Purchase Order
Add \$50 service fee if you require a prior invoice, or if you need to submit a purchase order or voucher, pay by wire transfer or EFT, or refund and reprocess a prior payment.

#### Options for Credit and Recognition

ACE CREDIT | College Credit
This course has been evaluated by the American Council on Education (ACE) and is recommended for the lower-division baccalaureate/associate degree, 3 semester hours in mathematics. Please note that the decision to accept specific credit recommendations is up to the academic institution accepting the credit.

#### Literacy, Accessibility, and Dyslexia

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Matrix Algebra