1: Introduction

Learning Objectives

After reading this chapter, you should be able to

1. define what a matrix is.
2. identify special types of matrices, and
3. identify when two matrices are equal.

What does a matrix look like?

Matrices are everywhere. If you have used a spreadsheet such as Excel or written numbers in a table, you have used a matrix. Matrices make presentation of numbers clearer and make calculations easier to program. Look at the matrix below about the sale of tires in a Blowoutr’us store – given by quarter and make of tires.

\[\begin{matrix} Tirestone\\ Michigan\\ Copper\\ \end{matrix} \stackrel{\mbox{Q1. Q2. Q3. Q4}}{\begin{bmatrix} 25 & 20 & 3 & 2 \\ 5 & 10 &15 &25 \\ 6 & 16 &7 & 27 \\ \end{bmatrix}} \nonumber \]

If one wants to know how many Copper tires were sold in Quarter 4, we go along the row Copper and column Q4 and find that it is 27.

So, What is a matrix?

A matrix is a rectangular array of elements. The elements can be symbolic expressions or/and numbers. Matrix \(\lbrack A\rbrack\) is denoted by

\[\lbrack A\rbrack = \begin{bmatrix} a_{11} & a_{12} & {.......} & a_{1n} \\ a_{21} & a_{22} & {.......} & a_{2n} \\ \vdots & & & \vdots \\ a_{m1} & a_{m2} & {.......} & a_{mn} \\ \end{bmatrix}\nonumber \]

Row \(i\) of \(\lbrack A\rbrack\) has \(n\) elements and is

\[\left\lbrack a_{i1}a_{i2}{....}a_{in} \right\rbrack \nonumber \]

and column \(j\) of \(\lbrack A\rbrack\) has \(m\) elements and is

\[\begin{bmatrix} a_{1j} \\ a_{2j} \\ \vdots \\ a_{mj} \\ \end{bmatrix} \nonumber \]

Each matrix has rows and columns and this defines the size of the matrix. If a matrix \(\lbrack A\rbrack\) has \(m\) rows and \(n\) columns, the size of the matrix is denoted by \(m \times n\). The matrix \(\lbrack A\rbrack\) may also be denoted by \(\lbrack A\rbrack_{m \times n}\) to show that \(\lbrack A\rbrack\) is a matrix with \(m\) rows and \(n\) columns.

Each entry in the matrix is called the entry or element of the matrix and is denoted by \(a_{ij}\) where \(i\) is the row number and \(j\) is the column number of the element.

The matrix for the tire sales example could be denoted by the matrix [A] as

\[\ \lbrack A\rbrack = \begin{bmatrix} 25 & 20 & 3 & 2 \\ 5 & 10 & 15 & 25 \\ 6 & 16 & 7 & 27 \\ \end{bmatrix} \nonumber \]

There are 3 rows and 4 columns, so the size of the matrix is \(3 \times 4\). In the above \(\lbrack A\rbrack\) matrix, \(a_{34} = 27\).

What are the special types of matrices?

Vector: A vector is a matrix that has only one row or one column. There are two types of vectors – row vectors and column vectors.

Row Vector:

If a matrix \(\lbrack B\rbrack\) has one row, it is called a row vector \(\lbrack B\rbrack = \lbrack b_{1} \;b_{2}\ldots\ldots b_{n}\rbrack\ \ \)and \(n\) is the dimension of the row vector.

Example 1

Give an example of a row vector.

Solution

\[\lbrack B\rbrack = \lbrack 25\ \ \ 20\ \ \ 3\ \ \ 2\ \ \ 0\rbrack\ \ \nonumber \]

is an example of a row vector of dimension 5.

Column vector:

If a matrix \(\lbrack C\rbrack\) has one column, it is called a column vector

\[\lbrack C\rbrack = \begin{bmatrix} c_{1} \\ \vdots \\ \vdots \\ c_{m} \\ \end{bmatrix} \nonumber \]

and \(m\) is the dimension of the vector.

Example 2

Give an example of a column vector.

Solution

\[\lbrack C\rbrack = \begin{bmatrix} 25 \\ 5 \\ 6 \\ \end{bmatrix} \nonumber \]

is an example of a column vector of dimension 3.

Submatrix:

If some row(s) or/and column(s) of a matrix \(\lbrack A\rbrack\) are deleted (no rows or columns may be deleted), the remaining matrix is called a submatrix of \(\lbrack A\rbrack\).

Example 3

Find some of the submatrices of the matrix

\[\lbrack A\rbrack = \begin{bmatrix} 4 & 6 & 2 \\ 3 & - 1 & 2 \\ \end{bmatrix} \nonumber \]

Solution

\[\begin{bmatrix} 4 & 6 & 2 \\ 3 & - 1 & 2 \\ \end{bmatrix},\ \ \begin{bmatrix} 4 & 6 \\ 3 & - 1 \\ \end{bmatrix},\ \ \begin{bmatrix} 4 & 6 & 2 \\ \end{bmatrix},\left\lbrack 4 \right\rbrack,\begin{bmatrix} 2 \\ 2 \\ \end{bmatrix} \nonumber \]

are some of the submatrices of \(\lbrack A\rbrack\). Can you find other submatrices of \(\lbrack A\rbrack\)?

Square matrix:

If the number of rows \(m\) of a matrix is equal to the number of columns \(n\) of a matrix \(\lbrack A\rbrack\), that is, \(m = n\), then \(\lbrack A\rbrack\) is called a square matrix. The entries \(a_{11},a_{22},...,a_{nn}\) are called the diagonal elements of a square matrix. Sometimes the diagonal of the matrix is also called the principal or main of the matrix.

Example 4

Give an example of a square matrix.

Solution

\[\lbrack A\rbrack = \begin{bmatrix} 25 & 20 & 3 \\ 5 & 10 & 15 \\ 6 & 15 & 7 \\ \end{bmatrix} \nonumber \]

is a square matrix as it has the same number of rows and columns, that is, 3. The diagonal elements of \(\lbrack A\rbrack\) are \(a_{11} = 25,\ \ a_{22} = 10,\ \ a_{33} = 7\).

Upper triangular matrix:

A \(n \times n\) matrix for which \(a_{ij} = 0,\ \ i > j\) for all \(i,j\) is called an upper triangular matrix. That is, all the elements below the diagonal entries are zero.

Example 5

Give an example of an upper triangular matrix.

Solution

\[\lbrack A\rbrack = \begin{bmatrix} 10 & - 7 & 0 \\ 0 & - 0.001 & 6 \\ 0 & 0 & 15005 \\ \end{bmatrix} \nonumber \]

is an upper triangular matrix.

Lower triangular matrix:

A \(n \times n\) matrix for which \(a_{ij} = 0,\ \ j > i\) for all \(i,j\) is called a lower triangular matrix. That is, all the elements above the diagonal entries are zero.

Example 6

Give an example of a lower triangular matrix.

Solution

\[\lbrack A\rbrack = \begin{bmatrix} 1 & 0 & 0 \\ 0.3 & 1 & 0 \\ 0.6 & 2.5 & 1 \\ \end{bmatrix} \nonumber \]

is a lower triangular matrix.

Diagonal matrix:

A square matrix with all non-diagonal elements equal to zero is called a diagonal matrix, that is, only the diagonal entries of the square matrix can be non-zero, (\(a_{ij} = 0,\ \ i \neq j\)).

Example 7

Give examples of a diagonal matrix.

Solution

\[\lbrack A\rbrack = \begin{bmatrix} 3 & 0 & 0 \\ 0 & 2.1 & 0 \\ 0 & 0 & 5 \\ \end{bmatrix} \nonumber \]

is a diagonal matrix.

Any or all the diagonal entries of a diagonal matrix can be zero. For example

\[\lbrack A\rbrack = \begin{bmatrix} 3 & 0 & 0 \\ 0 & 2.1 & 0 \\ 0 & 0 & 0 \\ \end{bmatrix} \nonumber \]

is also a diagonal matrix.

Identity matrix:

A diagonal matrix with all diagonal elements equal to 1 is called an identity matrix, (\(a_{ij} = 0,\ \ i \neq j\) for all \(i,j\) and \(a_{ii} = 1\) for all \(i\)).

Example 8

Give an example of an identity matrix.

Solution

\(\lbrack A\rbrack = \begin{bmatrix} 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \\ \end{bmatrix}\)

is an identity matrix.

Zero matrix:

A matrix whose all entries are zero is called a zero matrix, (\(a_{ij} = 0\) for all \(i\) and \(j\)).

Example 9

Give examples of a zero matrix.

Solution

\[\lbrack A\rbrack = \begin{bmatrix} 0 & 0 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & 0 \\ \end{bmatrix} \nonumber \]

\[\lbrack B\rbrack = \begin{bmatrix} 0 & 0 & 0 \\ 0&0&0 \\ \end{bmatrix} \nonumber \]

\[\lbrack C\rbrack = \begin{bmatrix} 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \\ \end{bmatrix} \nonumber \]

\[\lbrack D\rbrack = \begin{bmatrix} 0 & 0 & 0 \\ \end{bmatrix} \nonumber \]

are all examples of a zero matrix.

Tridiagonal matrices:

A tridiagonal matrix is a square matrix in which all elements not on the following are zero - the major diagonal, the diagonal above the major diagonal, and the diagonal below the major diagonal.

Example 10

Give an example of a tridiagonal matrix.

Solution

\[\lbrack A\rbrack = \begin{bmatrix} 2 & 4 & 0 & 0 \\ 2 & 3 & 9 & 0 \\ 0 & 0 & 5 & 2 \\ 0 & 0 & 3 & 6 \\ \end{bmatrix} \nonumber \]

is a tridiagonal matrix.

Do non-square matrices have diagonal entries?

Yes, for a \(m \times n\) matrix \(\lbrack A\rbrack\) , the diagonal entries are \(a_{11},a_{22}...,a_{k - 1,k - 1},a_{kk}\) where \(k = min\{ m,\ n\}\).

Example 11

What are the diagonal entries of

\[\lbrack A\rbrack = \begin{bmatrix} 3.2 & 5 \\ 6 & 7 \\ 2.9 & 3.2 \\ 5.6 & 7.8 \\ \end{bmatrix} \nonumber \]

Solution

The diagonal elements of \(\lbrack A\rbrack\) are \(a_{11} = 3.2\ and\ a_{22} = 7.\)

Diagonally Dominant Matrix:

A \(n \times n\) square matrix \([A]\) is a diagonally dominant matrix if

\[\left|a_{ii}\right| \geq \sum^n_{\substack{j=1\\ i \neq j}} \left|a_{ij}\right| \text{ for } i=1,2,....,n \nonumber\]

that is, for each row, the absolute value (also called magnitude) of the diagonal element is greater than or equal to the sum of the absolute values of the rest of the elements of that row.

Example 12

Give examples of matrices that are diagonally dominant and those that are not diagonally dominant.

Solution

1. The matrix

\[[A]=\begin{bmatrix}15 & 6 & 7\\ 2 & -4.1 & -2 \\ 3 & 2 & 6\end{bmatrix} \nonumber\]

is a diagonally dominant matrix.

Why? Because for each and every row, the answer to the question below is Yes.

Row 1: Is \(\left|a_{11}\right| \geq\left|a_{12}\right|+\left|a_{13}\right|\) ? Yes, because

\[\left|a_{11}\right|=|15|,\left|a_{12}\right|+\left|a_{13}\right|=|6|+|7|=13,15 \geq 13\nonumber\]

Row 2: Is \(\left|a_{22}\right| \geq\left|a_{21}\right|+\left|a_{23}\right|\) ? Yes, because

\[\left|a_{22}\right|=|-4.1|=4.1,\left|a_{21}\right|+\left|a_{23}\right|=|2|+|-2|=4,4.1 \geq 4\nonumber\]

Row 3: Is \(\left|a_{33}\right| \geq\left|a_{31}\right|+\left|a_{32}\right|\) ? Yes, because

\[\left|a_{33}\right|=|6|,\left|a_{31}\right|+\left|a_{32}\right|=|3|+|2|=5,6 \geq 5\nonumber\]

2. The matrix

\[[A] = \begin{bmatrix} -15 & 6 & 9 \\ 2 & -4 & -2 \\ 3 & -2 & 5 \end{bmatrix}\nonumber\]

is a diagonally dominant matrix.

Why? Because for each and every row, the answer to the question below is Yes.

Row 1: Is \(\left|a_{11}\right| \geq\left|a_{12}\right|+\left|a_{13}\right|\) ? Yes, because

\[\left|a_{11}\right|=|15|,\left|a_{12}\right|+\left|a_{13}\right|=|6|+|9|=15,15 \geq 15\nonumber\]

Row 2: Is \(\left|a_{22}\right| \geq\left|a_{21}\right|+\left|a_{23}\right|\) ? Yes, because

\[\left|a_{22}\right|=|-4|=4, \quad\left|a_{21}\right|+\left|a_{23}\right|=|2|+|-2|=4,4 \geq 4\nonumber\]

Row 3: Is \(\left|a_{33}\right| \geq\left|a_{31}\right|+\left|a_{32}\right|\)? Yes, because

\[\left|a_{33}\right|=|5|,\left|a_{31}\right|+\left|a_{32}\right|=|3|+|2|=5,5 \geq 5\nonumber\]

3. The matrix

\[[A] = \begin{bmatrix} -15 & 6 & 9 \\ 2 & -4.1 & -2 \\ 3 & -2 & 5 \end{bmatrix}\nonumber\]

is a diagonally dominant matrix.

Why? Because for each and every row, the answer to the question below is Yes.

Row 1: Is \(\left|a_{11}\right| \geq\left|a_{12}\right|+\left|a_{13}\right|\)? Yes, because

\[\left|a_{11}\right|=|15|,\left|a_{12}\right|+\left|a_{13}\right|=|6|+|9|=15,15 \geq 15\nonumber\]

Row 2: Is \(\left|a_{22}\right| \geq\left|a_{21}\right|+\left|a_{23}\right|\)? Yes, because

\[\left|a_{22}\right|=|-4.1|=4,\left|a_{21}\right|+\left|a_{23}\right|=|2|+|-2|=4,4.1 \geq 4\nonumber\]

Row 3: Is \(\left|a_{33}\right| \geq\left|a_{31}\right|+\left|a_{32}\right|\)? Yes, because

\[\left|a_{33}\right|=|5|,\left|a_{31}\right|+\left|a_{32}\right|=|3|+|2|=5,5 \geq 5\nonumber\]

4. The matrix

\[[A] = \begin{bmatrix} 25 & 5 & 1 \\ 64 & 8 & 1 \\ 144 & 12 & 1 \end{bmatrix}\nonumber\]

is not a diagonally dominant matrix.

Why? Because for each and every row, the answer to the question below is not a Yes.

Row 1: Is \(\left|a_{11}\right| \geq\left|a_{12}\right|+\left|a_{13}\right|\)? Yes, because

\[\left|a_{11}\right|=\left|25\right| ,\left|a_{12}\right|+\left|a_{13}\right|=|5|+|1|=6,25 \geq 6\nonumber\]

Row 2: Is \(\left|a_{22}\right| \geq\left|a_{21}\right|+\left|a_{23}\right|\)? No, because

\[\left|a_{22}\right|=|8|=8, \left|a_{21}\right|+\left|a_{23}\right|=|64|+|1|=65,8<65\nonumber\]

Row 3: Is \(\left|a_{33}\right| \geq\left|a_{31}\right|+\left|a_{32}\right|\)? No, because

\[\left|a_{33}\right|=|1|,\left|a_{31}\right|+\left|a_{32}\right|=|144|+|12|=156, 1<156\nonumber\]

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 (\(\geq\)).

Strictly diagonally dominant matrix:

A \(n \times n\) square matrix is a strictly diagonally dominant matrix if

\[\left|a_{ii}\right| > \sum^n_{\substack{j=1\\ i \neq j}} \left|a_{ij}\right| \text{ for } i=1,2,....,n\nonumber\]

that is, for each row, the absolute value of the diagonal element is strictly greater than the sum of the absolute values of the rest of the elements of that row.

Example 13

Give examples of strictly diagonally dominant matrices and not strictly diagonally dominant matrices.

Solution

1. The matrix

\[[A] = \begin{bmatrix} 15 & 6 & 7 \\ 2 & -4.1 & -2 \\ 3 & 2 & 6 \end{bmatrix}\nonumber\]

is a strictly diagonally dominant matrix

Why? Because for each and every row, the answer to the question below is Yes.

Row 1: Is \(\left|a_{11}\right| \geq\left|a_{12}\right|+\left|a_{13}\right|\)? Yes, because

\[\left|a_{11}\right|=|15|,\left|a_{12}\right|+\left|a_{13}\right|=|6|+|7|=13,15>13 .\nonumber\]

Row 2: Is \(\left|a_{22}\right| \geq\left|a_{21}\right|+\left|a_{23}\right|\)? Yes, because

\[\left|a_{22}\right|=|-4.1|=4.1,\left|a_{21}\right|+\left|a_{23}\right|=|2|+|-2|=4,4.1>4\nonumber\]

Row 3: Is \(\left|a_{33}\right| \geq\left|a_{31}\right|+\left|a_{32}\right|\)? Yes, because

\[\left|a_{33}\right|=|6|\left|a_{31}\right|+\left|a_{32}\right|=|3|+|2|=5,6>5\nonumber\] b) The matrix

\[[A]=\begin{bmatrix} 13 & 6 & 7 \\ 2 & -4.1 & -2 \\ 3 & 2 & 6 \end{bmatrix}\nonumber\]

is a not a strictly diagonally dominant matrix

Why? Because for each and every row, the answer to the question below is not a Yes.

Row 1: Is \(\left|a_{11}\right| \geq\left|a_{12}\right|+\left|a_{13}\right|\)? No, because

\[\left|a_{11}\right|=|13|,\left|a_{12}\right|+\left|a_{13}\right|=|6|+|7|=13,13 \ngtr 13\nonumber\]

Row 2: Is \(\left|a_{22}\right| \geq\left|a_{21}\right|+\left|a_{23}\right|\)? Yes, because

\[\left|a_{22}\right|=|-4.1|=4.1,\left|a_{21}\right|+\left|a_{23}\right|=|2|+|-2|=4,4.1>4\nonumber\]

Row 3: Is \(\left|a_{33}\right| \geq\left|a_{31}\right|+\left|a_{32}\right|\)? Yes, because

\[\left|a_{33}\right|=|6|,\left|a_{31}\right|+\left|a_{32}\right|=|3|+|2|=5,6>5\nonumber\]

3. The matrix

\[[A]=\begin{bmatrix} 25 & 5 & 1 \\ 64 & 8 & 1 \\ 144 & 12 & 1 \end{bmatrix}\nonumber\]

is not a strictly diagonally dominant matrix.

Why? Because for each and every row, the answer to the question below is not a Yes.

Row 1: Is \(\left|a_{11}\right| \geq\left|a_{12}\right|+\left|a_{13}\right|\)? Yes, because

\[\left|a_{11}\right|=|25|,\left|a_{12}\right|+\left|a_{13}\right|=|5|+|1|=6,25>6\nonumber\]

Row 2: Is \(\left|a_{22}\right| \geq\left|a_{21}\right|+\left|a_{23}\right|\)? No, because

\[\left|a_{22}\right|=|8|=8, \left|a_{21}\right|+\left|a_{23}\right|=|64|+|1|=65,8<65\nonumber\]

Row 3: Is \(\left|a_{33}\right| \geq\left|a_{31}\right|+\left|a_{32}\right|\)? No, because

\[\left|a_{33}\right|=|1||| a_{31}|+| a_{32}|=| 144|+| 12 \mid=156, 1<156\nonumber\]

Irreducible diagonally dominant matrix

A \(n \times n\) square matrix is an irreducible diagonally dominant matrix if

\[[A]\ \text{is irreducible},\nonumber\]

\[\left|a_{ii}\right| \geq \sum^n_{\substack{j=1\\ i \neq j}} \left|a_{ij}\right| \text{ for } i=1,2,....,n\ \text{and}\nonumber\]

\[\left|a_{ii}\right| > \sum^n_{\substack{j=1\\ i \neq j}} \left|a_{ij}\right| \text{ for at least one row, } i=1,2,....,n\nonumber\]

The second condition means that for each row, the absolute value (also called magnitude) of the diagonal element is greater than or equal to the sum of the absolute values of the rest of the elements of that row. The third condition means that for at least one row, the absolute value (also called magnitude) of the diagonal element is greater than the sum of the absolute values of the rest of the elements of that row.

Example 14

Give examples of matrices that are irreducibly diagonally dominant and those that are not irreducibly diagonally dominant.

Solution

1. The matrix

\[[A]=\begin{bmatrix} 15 & 6 & 7 \\ 2 & -4.1 & -2 \\ 3 & 2 & 6 \end{bmatrix}\nonumber\]

is an irreducible diagonally dominant matrix.

Why? Because the answer to every question below is Yes.

Is \([A]\) irreducible? Yes.

Row 1: Is \(\left|a_{11}\right| \geq\left|a_{12}\right|+\left|a_{13}\right|\)? Yes, because

\[\left|a_{11}\right|=|15|,\left|a_{12}\right|+\left|a_{13}\right|=|6|+|7|=13,15>13 .\nonumber\]

Row 2: Is \(\left|a_{22}\right| \geq\left|a_{21}\right|+\left|a_{23}\right|\)? Yes, because

\[\left|a_{22}\right|=|-4.1|=4.1,\left|a_{21}\right|+\left|a_{23}\right|=|2|+|-2|=4,4.1>4\nonumber\]

Row 3: Is \(\left|a_{33}\right| \geq\left|a_{31}\right|+\left|a_{32}\right|\)? Yes, because

\[\left|a_{33}\right|=|6|\left|a_{31}\right|+\left|a_{32}\right|=|3|+|2|=5,6>5\nonumber\]

Is the inequality satisfied strictly for at least one row? Yes, it is satisfied for Rows 1, 2 and 3.

2. The matrix

\[[A]=\begin{bmatrix} -15 & 6 & 9 \\ 2 & -4 & -2 \\ 3 & -2 & 5 \end{bmatrix}\nonumber\]

is a not an irreducible diagonally dominant matrix.

Why? Because the answer to every question below is not a Yes.

Is \([A]\) irreducible? Yes.

Row 1: Is \(\left|a_{11}\right| \geq\left|a_{12}\right|+\left|a_{13}\right|\)? Yes, because

\[\left|a_{11}\right|=|15|,\left|a_{12}\right|+\left|a_{13}\right|=|6|+|9|=15,15 \geq 15\nonumber\]

Row 2: Is \(\left|a_{22}\right| \geq\left|a_{21}\right|+\left|a_{23}\right|\)? Yes, because

\[\left|a_{22}\right|=|-4|=4, \left|a_{21}\right|+\left|a_{23}\right|=|2|+|-2|=4,4 \geq 4\nonumber\]

Row 3: Is \(\left|a_{33}\right| \geq\left|a_{31}\right|+\left|a_{32}\right|\)? Yes, because

\[\left|a_{33}\right|=|5|,\left|a_{31}\right|+\left|a_{32}\right|=|3|+|2|=5,5 \geq 5\nonumber\]

Is the inequality satisfied strictly for at least one row? No.

3. The matrix

\[[A]=\begin{bmatrix} -15 & 6 & 9 \\ 2 & -4.1 & -2 \\ 3 & -2 & 5 \end{bmatrix}\nonumber\]

is an irreducible diagonally dominant matrix.

Why? Because the answer to every question below is Yes.

Is \([A]\) irreducible? Yes.

Row 1: Is \(\left|a_{11}\right| \geq\left|a_{12}\right|+\left|a_{13}\right|\)? Yes, because

\[\left|a_{11}\right|=|15|,\left|a_{12}\right|+\left|a_{13}\right|=|6|+|9|=15,15 \geq 15\nonumber\]

Row 2: Is \(\left|a_{22}\right| \geq\left|a_{21}\right|+\left|a_{23}\right|\)? Yes, because

\[\left|a_{22}\right|=|-4.1|=4.1, \left|a_{21}\right|+\left|a_{23}\right|=|2|+|-2|=4,4.1 \geq 4\nonumber\]

Row 3: Is \(\left|a_{33}\right| \geq\left|a_{31}\right|+\left|a_{32}\right|\)? Yes, because

\[\left|a_{33}\right|=|5|,\left|a_{31}\right|+\left|a_{32}\right|=|3|+|2|=5,5 \geq 5\nonumber\]

Is the inequality satisfied strictly for at least one row? Yes, it is satisfied for Row 2.

4. The matrix

\[[A]=\begin{bmatrix} 25 & 5 & 1 \\ 64 & 8 & 1 \\ 144 & 12 & 1 \end{bmatrix}\nonumber\]

is not an irreducible diagonally dominant matrix.

Why? Because the answer to every question below is not a Yes.

Is \([A]\) irreducible? Yes.

Row 1: Is \(\left|a_{11}\right| \geq\left|a_{12}\right|+\left|a_{13}\right|\)? Yes, because

\[\left|a_{11}\right|=|25|,\left|a_{12}\right|+\left|a_{13}\right|=|5|+|1|=6,25>6\nonumber\]

Row 2: Is \(\left|a_{22}\right| \geq\left|a_{21}\right|+\left|a_{23}\right|\)? No, because

\[\left|a_{22}\right|=|8|=8, \left|a_{21}\right|+\left|a_{23}\right|=|64|+|1|=65,8<65\nonumber\]

Row 3: Is \(\left|a_{33}\right| \geq\left|a_{31}\right|+\left|a_{32}\right|\)? No, because

\[\left|a_{33}\right|=|1||| a_{31}|+| a_{32}|=| 144|+| 12 \mid=156, 1<156\nonumber\]

There is no need to check for strict inequality condition..

Irreducible matrix:

A square matrix is called reducible matrix if the following is true. Take the indices \(i=1,2,....,n\) and see if they can be divided into two disjoint nonempty sets \(i_1,i_2,....,i_\alpha\) and \(j_1,j_2,....,j_\beta\) such that

\[n=\alpha + \beta,\ \text{and}\nonumber\]

and

\[a_{i_k j_l}=0,\ k=1,2,....,\alpha\ \text{and}\ l=1,2,....,\beta\nonumber\]

If the square matrix is not reducible, it is called an irreducible matrix.

A square matrix \([A]\) is called reducible matrix if and only if for any perturbation matrix \([P]\), the matrix multiplication \([P]^T[A][P]\) results in a block upper triangular matrix.

Example 15

Give examples of irreducible and reducible matrices.

Solution

1. The matrix

\[\begin{bmatrix} 0 & 5 & 7 \\ 8 & 0 & 0 \\ 10 & 0 & 0 \end{bmatrix}\nonumber\]

is an irreducible matrix.

2. The matrix

\[\begin{bmatrix} 5 & 0 & 0 \\ 0 & 4 & 6 \\ 10 & 0 & 0 \end{bmatrix}\nonumber\]

is a reducible matrix. Why? Take the indices \(i=1,2,3\) and see that they can be divided into two disjoint nonempty sets 1 and 2,3 such that,

\[\alpha = 1, \beta = 2,\ \text{giving}\ \alpha + \beta = 1+2 = 3,\ \text{and}\nonumber\]

\[a_{i_k j_l}=0,\ k=1 \ \text{and}\ l = 1,2\nonumber\]

Consequences of diagonally dominant matrices

If a square matrix is strictly diagonally dominant

• then the matrix is non-singular.

• then if the matrix is symmetric with non-negative diagonal entries, the matrix is positive semi-definite.

• then if the matrix is the coefficient matrix for a set of simultaneous linear equations, the iterative Gauss-Seidel numerical method will always converge.

• then if the matrix is the coefficient matrix for a set of simultaneous linear equations, the iterative Jordan numerical method will always converge.

• then if the diagonal entries of the matrix are positive, the real parts of the matrix eigenvalues are positive.

• then if the diagonal entries of the matrix are negative, the real parts of the matrix eigenvalues are negative.

• then if the matrix is column dominant, no pivoting is needed for Gaussian elimination.

• then if the matrix is column dominant, no pivoting is needed for LU factorization.

If a square matrix is irreducible diagonally dominant

• then if the matrix is the coefficient matrix for a set of simultaneous linear equations, the iterative Gauss-Seidel numerical method will always converge.

• then if the matrix is the coefficient matrix for a set of simultaneous linear equations, the iterative Jordan numerical method will always converge.

• the matrix is non-singular.

If a square matrix is diagonally dominant (also called weakly diagonally dominant)

• then if the matrix is column dominant, no pivoting is needed for Gaussian elimination.

• then if the matrix is column dominant, no pivoting is needed for LU factorization.

Equal matrices:

Two matrices [A] and [B] are equal if the size of [A] and [B] is the same (number of rows and columns of [A] are same as that of [B]) and \(a_{ij} = b_{ij}\) for all i and j.

Example 16

What would make

\[[A] = \begin{bmatrix} 2 & 3 \\ 6 & 7 \\ \end{bmatrix}\nonumber\]

to be equal to

\[[B] = \begin{bmatrix} b_{11} & 3 \\ 6 & b_{22} \\ \end{bmatrix}\nonumber\]

Solution

The two matrices \([A]\)and \([B]\) would be equal if \(b_{11} = 2\) and \(b_{22} = 7\).

Key Terms:

Matrix

Vector

Submatrix

Square matrix

Equal matrices

Zero matrix

Identity matrix

Diagonal matrix

Upper triangular matrix

Lower triangular matrix

Tri-diagonal matrix

Introduction Quiz

Quiz 1

For an \(n \times n\) upper triangular matrix \(\left\lbrack A \right\rbrack\),

(A) \(a_{ij} = 0,i > j\)

(B) \(a_{ij} = 0,j > i\)

(C) \(a_{ij} \neq 0,i > j\)

(D) \(a_{ij} \neq 0,j > i\)

Quiz 2

Which one of these square matrices is strictly diagonally dominant?

(A) \(\begin{bmatrix} 5 & 7 & 0 \\ 3 & - 6 & 2 \\ 2 & 2 & 9 \\ \end{bmatrix}\)

(B) \(\begin{bmatrix} 7 & - 5 & - 2 \\ 6 & - 13 & - 7 \\ 6 & - 7 & - 13 \\ \end{bmatrix}\)

(C) \(\begin{bmatrix} 8 & - 5 & - 2 \\ 6 & - 14 & - 7 \\ 6 & - 7 & - 13 \\ \end{bmatrix}\)

(D) \(\begin{bmatrix} 8 & 5 & 2 \\ 6 & 14 & 7 \\ 6 & 7.5 & 14 \\ \end{bmatrix}\)

Quiz 3

The order of the following matrix is

\[\begin{bmatrix} 4 & - 6 & - 7 & 2 \\ 3 & 2 & - 5 & 6 \\ \end{bmatrix} \nonumber \]

(A) \(4 \times 2\)

(B) \(2 \times 4\)

(C) \(8 \times 1\)

(D) not defined

Quiz 4

To make the following two matrices equal

\[\left\lbrack A \right\rbrack = \begin{bmatrix} 5 & - 6 & 7 \\ 3 & 2 & 5 \\ \end{bmatrix} \nonumber \]

\[\left\lbrack B \right\rbrack = \begin{bmatrix} 5 & p & 7 \\ 3 & 2 & 5 \\ \end{bmatrix} \nonumber \] the value of \(p\) is

(A) \(- 6\)

(B) \(6\)

(C) \(0\)

(D) \(7\)

Quiz 5

For a square \(n \times n\) matrix \(\left\lbrack A \right\rbrack\) to be an identity matrix,

(A) \(a_{ij} \neq 0,i = j;a_{ij} = 0,i = j\)

(B) \(a_{ij} = 0,i \neq j;a_{ij} = 1,i = j\)

(C) \(a_{ij} = 0,i \neq j;a_{ij} = i,i = j\)

(D) \(a_{ij} = 0,i \neq j;a_{ij} > 0,i = j\)

Quiz 6

To make the following square matrix to be diagonally dominant, the value of \(p\)needs to be

\[\begin{bmatrix} 6 & - 2 & - 4 \\ 7 & 9 & 1 \\ 8 & - 5 & p \\ \end{bmatrix} \nonumber \]

(A) greater than or equal to 13

(B) greater than 3

(C) greater than or equal to 3

(D) greater than 13

Introduction Exercise

Exercise 1

Write an example of a row vector of dimension 4.

Answer

\(\begin{bmatrix} 5 & 6 & 2 & 3 \\ \end{bmatrix}\)

Exercise 2

Write an example of a column vector of dimension 4.

Answer

\(\ \begin{bmatrix} 5 \\ - 7 \\ 3 \\ 2.5 \\ \end{bmatrix}\)

Exercise 3

Write an example of a square matrix of order \(4 \times 4\).

Answer

\(\ \begin{bmatrix} 9 & 0 & - 2 & 3 \\ - 2 & 3 & 5 & 1 \\ 1.5 & 6 & 7 & 8 \\ 1.1 & 2 & 3 & 4 \\ \end{bmatrix}\)

Exercise 4

Write an example of a tri-diagonal matrix of order \(4 \times 4\).

Answer

\(\ \begin{bmatrix} 6 & 3 & 0 & 0 \\ 2.1 & 2 & 2.2 & 0 \\ 0 & 6.2 & - 3 & 3.5 \\ 0 & 0 & 2.1 & 4.1 \\ \end{bmatrix}\)

Exercise 5

Write an example of a identity matrix of order \(5 \times 5\).

Answer

\(\ \begin{bmatrix} 1 & 0 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 & 0 \\ 0 & 0 & 1 & 0 & 0 \\ 0 & 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 0 & 1 \\ \end{bmatrix}\)

Exercise 6

Write an example of a upper triangular matrix of order \(4 \times 4\).

Answer

\(\ \begin{bmatrix} 6 & 2 & 3 & 9 \\ 0 & 1 & 2 & 3 \\ 0 & 0 & 4 & 5 \\ 0 & 0 & 0 & 6 \\ \end{bmatrix}\)

Exercise 7

Write an example of a lower triangular matrix of order \(4 \times 4\).

Answer

\(\ \begin{bmatrix} 2 & 0 & 0 & 0 \\ 3 & 1 & 0 & 0 \\ 4 & 2 & 4 & 0 \\ 5 & 3 & 5 & 6 \\ \end{bmatrix}\)

Exercise 8

Which of these matrices are strictly diagonally dominant?

1. \(\left\lbrack A \right\rbrack = \begin{bmatrix} 15 & 6 & 7 \\ 2 & - 4 & 2 \\ 3 & 2 & 6 \\ \end{bmatrix}\)
2. \(\left\lbrack A \right\rbrack = \begin{bmatrix} 5 & 6 & 7 \\ 2 & - 4 & 2 \\ 3 & 2 & - 5 \\ \end{bmatrix}\)
3. \(\left\lbrack A \right\rbrack = \begin{bmatrix} 5 & 3 & 2 \\ 6 & - 8 & 2 \\ 7 & - 5 & 12 \\ \end{bmatrix}\)

Answer

(A) Yes (B) No (C) No

Exercise 9

Find all the submatrices of

\[\lbrack A\rbrack = \begin{bmatrix} 10 & - 7 & 0 \\ 0 & - 0.001 & 6 \\ \end{bmatrix} \nonumber \]

Answer

\(\left\lbrack 10 \right\rbrack\) \(\left\lbrack - 7 \right\rbrack\) , \(\left\lbrack 0 \right\rbrack\), \(\left\lbrack - 0.001 \right\rbrack\), \(\left\lbrack 6 \right\rbrack\) \(\begin{bmatrix} 10 \\ 0 \\ \end{bmatrix}\), \(\begin{bmatrix} - 7 \\ - .001 \\ \end{bmatrix}\), \(\begin{bmatrix} 0 \\ 6 \\ \end{bmatrix}\), \(\begin{bmatrix} 10 & - 7 & 0 \\ \end{bmatrix}\), \(\begin{bmatrix} 0 & - 0.001 & 6 \\ \end{bmatrix}\), \(\begin{bmatrix} 10 & - 7 \\ 0 & - 0.001 \\ \end{bmatrix}\),\(\begin{bmatrix} 10 & 0 \\ 0 & 6 \\ \end{bmatrix}\), \(\begin{bmatrix} - 7 & 0 \\ - 0.001 & 6 \\ \end{bmatrix}\), \(\left\lbrack 10, - 7 \right\rbrack\) , \(\left\lbrack 10,0 \right\rbrack\), \(\left\lbrack - 7,0 \right\rbrack\) , \(\left\lbrack 0,6 \right\rbrack\) , \(\left\lbrack 0, - 0.001 \right\rbrack\), \(\left\lbrack - 0.001,6 \right\rbrack\).

Exercise 10

If

\[\lbrack A\rbrack = \begin{bmatrix} 4 & - 1 \\ 0 & 2 \\ \end{bmatrix}, \nonumber \]

what are \(b_{11}\) and \(b_{12}\) in

\[\lbrack B\rbrack = \begin{bmatrix} b_{11} & b_{12} \\ 0 & 4 \\ \end{bmatrix} \nonumber \]

if \(\lbrack B\rbrack = 2\lbrack A\rbrack\).

Answer

\(\ 8, - 2\)

Exercise 11

Are matrix

\[\lbrack A\rbrack = \begin{bmatrix} 10 & - 7 & 0 \\ 0 & - 0.001 & 6 \\ \end{bmatrix} \nonumber \]

and matrix

\[\lbrack B\rbrack = \begin{bmatrix} 10 & 0 \\ - 7 & - 0.001 \\ 0 & 6 \\ \end{bmatrix} \nonumber \]

equal?

Answer

No

Exercise 12

A square matrix \(\lbrack A\rbrack\) is lower triangular if

1. \(a_{ij} = 0\) for \(i > j\)
2. \(a_{ij} = 0\) for \(j > i\)
3. \(a_{ij} = 0\) for \(i = j\)
4. \(a_{ij} = 0\) for \(i + j = odd\ integer\)

Answer

B

Exercise 13

A square matrix \(\lbrack A\rbrack\) is upper triangular if

1. \(a_{ij} = 0\) for \(i > j\)
2. \(a_{ij} = 0\) for \(j > i\)
3. \(a_{ij} = 0\) for \(i = j\)
4. \(a_{ij} = 0\) for \(i + j = odd\ integer\)

Answer

A