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 


This post is brought to you by

Advertisements

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.

______________________________________________

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.

Is a square matrix diagonal or not?

A square matrix A is diagonal if all the elements on the off-diagonal are zero. That is, A(i,j)=0 for i~=j.

In this posting, I show a MATLAB program that finds whether a square matrix is diagonal by using three different methods. These are academic ways to reinforce programming skills in a student.

The MATLAB program can be downloaded as a Mfile (better to download it, as single quotes from the web-post do not translate correctly with the MATLAB editor). The html file showing the mfile and the command window output is also available.

%% IS A GIVEN SQUARE MATRIX A DIAGONAL MATRIX?
% Language : Matlab 2007a
% Authors : Autar Kaw
% Last Revised : November 15, 2008
% Abstract: This program shows you three ways of finding out
% if a square matrix is a diagonal matrix. A square matrix is
% diagonal if all the off-diagonal elements are zero, that is
% A(i,j)=0 for i~=j.
clc
clear all
disp(‘This program shows you three ways of finding out’)
disp(‘if a square matrix is a diagonal matrix.’)
disp(‘A square matrix is diagonal if all the off-diagonal’)
disp(‘elements are zero, that is A(i,j)=0 for i~=j.’)
disp(‘ ‘)
%% INPUTS
% The square matrix
A=[1 0 0 0;0 3.4 0 0; 0 0 -4.5 0;0 0 0 0];
disp (‘INPUTS’)
disp(‘Here is the square matrix’)
A
disp(‘ ‘)

%% FIRST SOLUTION
% This is based on counting the number of zeros on
% off the diagonal. If this count is n^2-n then it
% is a diagonal matrix, otherwise it is not a diagonal matrix

%size gives how many rows and columns in the A matrix
rowcol=size(A);
n=rowcol(1);
% count = how many zeros not on the diagonal
count=0;
for i=1:1:n
for j=1:1:n
if A(i,j)==0 & i~=j
count=count+1;
end
end
end
disp(‘FIRST WAY’)
if count==n^2-n
disp(‘Matrix is diagonal’)
else
disp(‘Matrix is NOT diagonal’)
end

%% SECOND SOLUTION
% This is based on finding if any of the off-diagonal elements
% are nozero. As soon as this condition is met, the matrix can be
% deemed not diagonal. If the condition is never met, the matrix is
% diagonal

%size gives how many rows and columns in the A matrix
rowcol=size(A);
n=rowcol(1);
% flag = keeps track if it is diagonal or not
% flag = 1 if matrix is diagonal
% flag = 2 if matrix is not diagonal

% Assuming matrix is diagonal
flag=1;
for i=1:1:n
for j=1:1:n
% flag=2 if off-diagonal element is nonzero.
if A(i,j)~=0 & i~=j
flag=2;
end
end
end
disp(‘ ‘)
disp(‘SECOND WAY’)
if flag==1
disp(‘Matrix is diagonal’)
else
disp(‘Matrix is NOT diagonal’)
end

%% THIRD SOLUTION
% This is based on finding if the sum of the absolute value of
% the off-diagonal elements is nonzero.
% If the sum is nonzero, the matrix is NOT diagonal.
% If the sum is zero, the matrix is diagonal

%size gives how many rows and columns in the A matrix
rowcol=size(A);
n=rowcol(1);

% sum_off_diagonal= sum of absolute value of off-diagonal elements
sum_off_diagonal=0;
for i=1:1:n
for j=1:1:n
if i~=j
sum_off_diagonal=sum_off_diagonal+abs(A(i,j));
end
end
end

disp(‘ ‘)
disp(‘THIRD WAY’)
if sum_off_diagonal==0
disp(‘Matrix is diagonal’)
else
disp(‘Matrix is NOT diagonal’)
end

This post is brought to you by Holistic Numerical Methods: Numerical Methods for the STEM undergraduate at http://numericalmethods.eng.usf.edu.

An abridged (for low cost) book on Numerical Methods with Applications will be in print (includes problem sets, TOC, index) on December 10, 2008 and available at lulu storefront.

Subscribe to the blog via a reader or email to stay updated with this blog. Let the information follow you.