### Matrix Algebra

Matrix Algebra Review

taught by Robert LaBudde

Aim of Course:

Statistics deals with collections of data organized in 1, 2, 3 or more dimensions. Compactly representing such data is best accomplished by the use of matrix notation, particularly when solutions to optimization (e.g., regression) or estimating (i.e, models) are involved. This online course, "Matrix Algebra Review" will provide 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. After successfully completing this course, you will be able to use and understand vector and matrix operations and equations, find and use a matrix inverse, and use and understand the eigenset of a symmetric matrix.

Vector and matrix notation: An excerpt from the course materials is available here.

This course may be taken individually (one-off) or as part of a certificate program.

Course Program:

## 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
- Quadratic forms
- Applications to statistics

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.

# Matrix Algebra Review

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.

Level:

Intermediate

You should be familiar with introductory statistics. Try these self tests to check your knowledge.

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.

Organization of the Course:

This course takes place online at the Institute for 4 weeks. During each course week, you participate at times of your own choosing - there are no set times when you must be online. Course participants will be given access to a private discussion board. In class discussions led by the instructor, you can post questions, seek clarification, and interact with your fellow students and the instructor.

At the beginning of each week, you receive the relevant material, in addition to answers to exercises from the previous session. During the week, you are expected to go over the course materials, work through exercises, and submit answers. Discussion among participants is encouraged. The instructor will provide answers and comments, and at the end of the week, you will receive individual feedback on your homework answers.

Time Requirement:

About 15 hours per week, at times of your choosing.

Credit:

Students come to the Institute for a variety of reasons. As you begin the course, you will be asked to specify your category:

- You may be interested only in learning the material presented, and not be concerned with grades or a record of completion.
- You may be enrolled in PASS (Programs in Analytics and Statistical Studies) that requires demonstration of proficiency in the subject, in which case your work will be assessed for a grade.
- You may require a "Record of Course Completion," along with professional development credit in the form of Continuing Education Units (CEU's). For those successfully completing the course, CEU's and a record of course completion will be issued by The Institute, upon request.

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, and 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.

# Matrix Algebra Review

March 17, 2017 to April 14, 2017June 23, 2017 to July 21, 2017March 16, 2018 to April 13, 2018