Category: Matrix Algebra

Matrix Algebra: Binary Operations

Many university STEM major programs have reduced the credit hours for a course in Matrix Algebra or have simply dropped the course from their curriculum.   The content of Matrix Algebra in many cases is taught just in time where needed.  This approach can leave a student with many conceptual holes in the required knowledge of matrix algebra.

In this series of blogs, we bring to you ten topics that are of immediate and intermediate interest for Matrix Algebra.

Here is the third topic where we talk about binary operations of matrices – subtraction, addition, and multiplication.  Linear combination of matrices and rules of binary operations are discussed. Get all the resources in form of textbook content, lecture videos, multiple choice test, problem set, and PowerPoint presentation.

Binary Operations

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Matrix Algebra: Introduction

Many university STEM major programs have reduced the credit hours for a course in Matrix Algebra or have simply dropped the course from their curriculum.   The content of Matrix Algebra in many cases is taught just in time where needed.  This approach can leave a student with many conceptual holes in the required knowledge of matrix algebra.

In this series of blogs, we bring to you ten topics that are of immediate and intermediate interest for Matrix Algebra.

Here is the first topic where we define a matrix, vector, submatrix, square matrix, triangular matrix (upper and lower), diagonal matrix, identity matrix, and diagonally dominant matrix.  Get the information in form of textbook content, lecture videos, multiple choice test, problem set and PowerPoint presentation.

Introduction 

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How much computational time does it take to find the inverse of a square matrix using Gauss Jordan method?  Part 1 of 2.

Problem Statement

How much computational time does it take to find the inverse of a square matrix using Gauss Jordan method?  Part 1 of 2.

Solution

To understand the solution, you should be familiar with the Gauss Jordan method of finding the inverse of a square matrix.  Peter Young of UCSC describes it briefly in this pdf file while if you like watching an example via a video, you can see PatrickJMT doing so.  You also need to read a previous blog where we calculated the computational time needed for the forward elimination steps on a square matrix in the Naïve Gauss elimination method.   We are now ready to estimate the computational time required for Gauss Jordan method of finding the inverse of a square matrix.

GJ Inverse Blog

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Clearing up the confusion about diagonally dominant matrices – Part 3

You can view the above document as a pdf file as well.

Other blogs on diagonally dominant matrices
Clearing up the confusion about diagonally dominant matrices – Part 1

Clearing up the confusion about diagonally dominant matrices – Part 2

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Clearing up the confusion about diagonally dominant matrices – Part 2

In a previous post, we discussed the confusion about the definition and associated properties of diagonally dominant matrices.  In this blog, we answer the next question.

What is a weak diagonally dominant matrix?

The answer is simple – the definition of a weak(ly) diagonally dominant matrix is identical to that of a diagonally dominant matrix as the inequality used for the check is a weak inequality of greater than or equal to (≥).  See the previous post on Clearing up the confusion about diagonally dominant matrices – Part 1 where we define a diagonally dominant matrix.

Other blogs on diagonally dominant matrices
Clearing up the confusion about diagonally dominant matrices – Part 1

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Misconceptions about diagonal and tridiagonal matrices

A reader wrote: “I purchased on lulu the 2nd edition of your Introduction to Matrix Algebra for self study, and the book just arrived. I started reading it and found some annoying errors. For example on Chapter 1, page 5: for the first (diagonal) matrix, why is there a zero located in a33, when you defined on the previous page that only diagonal entries of square matrix can be non-zero (this answer is different on your free online pdf) . Also, on page 6 for the tridiagonal matrix, why is there a zero located in the diagonal below the major diagonal? I was wondering if you can provide me with the list of errors and corrections, because it’s going to be very difficult to study the material on my own and the errors in the book just makes it more frustrating.”

My answer: “There is no erratum issued yet on the book.

A diagonal matrix is diagonal based on the nondiagonal elements being zero. The diagonal elements have no restrictions. They can be zero or nonzero.

A tridiagonal matrix is a square matrix in which all elements not on the following are zero – the major diagonal, the above the major diagonal, and the diagonal below the major diagonal. The major diagonal, the diagonal above the major diagonal, and the diagonal below the major diagonal have no restrictions. They can be zero or nonzero.

The concerns you have raised are some of the common misconceptions students develop about these special matrices.

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This post is brought to you by Holistic Numerical Methods: Numerical Methods for the STEM undergraduate at http://nm.MathForCollege.com, the textbook on Numerical Methods with Applications available from the lulu storefront, the textbook on Introduction to Programming Concepts Using MATLAB, and the YouTube video lectures available at http://nm.MathForCollege.com/videos. Subscribe to the blog via a reader or email to stay updated with this blog. Let the information follow you.