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Chapter 04.01 MATRICES 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. Q1 Q2 Q3 Q4 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. 04.01.1

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Introduction to matrix algebra: General Engineering

04.01.8

Chapter 04.01Introduction 04.01.7

Chapter 04.01

MATRICESAfter reading this chapter, you should be able to

1. matrix" define what a matrix is.2. identify special types of matrices, and

3. matrices" identify when two matrices are equal.

What does a matrix look like XE "matrix" ?

Matrices XE "Matrices" are everywhere. If you have used a spreadsheet such as Excel or written numbers in a table, you have used a matrix XE "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 Blowoutrus store given by quarter and make of tires.

Q1 Q2 Q3 Q4

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 XE "matrix" ?

A matrix XE "matrix" is a rectangular array of elements. The elements can be symbolic expressions or/and numbers. Matrix XE "Matrix" is denoted by

Row of has elements and is

and column of has elements and is

Each matrix XE "matrix" has rows and columns and this defines the size of the matrix. If a matrix has rows and columns, the size of the matrix is denoted by . The matrix may also be denoted by to show that is a matrix with rows and columns.

Each entry XE "entry" in the matrix XE "matrix" is called the entry or element of the matrix XE "element of the matrix" and is denoted by where is the row number and is the column number of the element.

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

.

There are 3 rows and 4 columns, so the size of the matrix XE "matrix" is . In the above matrix, .

What are the special types of matrices XE "special types of matrices" ?

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

XE "vector" :

If a matrix has one row, it is called a row vector XE "row vector" and is the dimension of the row vector XE "row vector" .Example 1

Give an example of a row vector XE "row vector" .

Solution

is an example of a row vector XE "row vector" of dimension 5.

Column vector XE "Column vector"

XE "vector" :

If a matrix XE "matrix" has one column, it is called a column vector XE "column vector"

and is the dimension of the vector XE "vector" .

Example 2

Give an example of a column vector XE "column vector" .

Solution

is an example of a column vector XE "column vector" of dimension 3.Submatrix XE "Submatrix" :

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

Example 3

Find some of the submatrices of the matrix

Solution

are some of the submatrices of . Can you find other submatrices of ?

Square matrix XE "Square matrix"

XE "matrix" :

If the number of rows of a matrix is equal to the number of columns of a matrix , that is, , then is called a square matrix XE "square matrix" . The entries are called the diagonal XE "diagonal" elements XE "diagonal elements" of a square matrix. Sometimes the diagonal of the matrix is also called the principal XE "principal" or main of the matrix.

Example 4

Give an example of a square matrix XE "square matrix" .

Solution

is a square matrix XE "square matrix" as it has the same number of rows and columns, that is, 3. The diagonal XE "diagonal" elements XE "diagonal elements" of are .

Upper triangular matrix XE "Upper triangular matrix"

XE "matrix" :

A matrix for which for all is called an upper triangular matrix XE "upper triangular matrix" . That is, all the elements below the diagonal XE "diagonal" entries are zero.Example 5

Give an example of an upper triangular matrix XE "upper triangular matrix" .Solution

is an upper triangular matrix XE "upper triangular matrix" .

Lower triangular matrix XE "Lower triangular matrix"

XE "matrix" :

A matrix for which for all is called a lower triangular matrix XE "lower triangular matrix" . That is, all the elements above the diagonal XE "diagonal" entries are zero.Example 6

Give an example of a lower triangular matrix XE "lower triangular matrix" .

Solution

is a lower triangular matrix XE "lower triangular matrix" .

Diagonal XE "Diagonal" matrix XE "Diagonal matrix"

XE "matrix" :

A square matrix XE "square matrix" with all non-diagonal XE "diagonal" elements XE "diagonal elements" equal to zero is called a diagonal matrix XE "diagonal matrix" , that is, only the diagonal entries of the square matrix can be non-zero, ().

Example 7

Give examples of a diagonal XE "diagonal" matrix XE "diagonal matrix" .

Solution

is a diagonal XE "diagonal" matrix XE "diagonal matrix" .

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

is also a diagonal XE "diagonal" matrix XE "diagonal matrix" .

Identity matrix XE "Identity matrix"

XE "matrix" :

A diagonal XE "diagonal" matrix XE "diagonal matrix" with all diagonal elements XE "diagonal elements" equal to 1 is called an identity matrix XE "identity matrix" , (for all and for all ).

Example 8

Give an example of an identity matrix XE "identity matrix" .

Solution

is an identity matrix XE "identity matrix" .

Zero matrix XE "Zero matrix"

XE "matrix" :

A matrix whose all entries are zero is called a zero matrix XE "zero matrix" , ( for all and).

Example 9

Give examples of a zero matrix XE "zero matrix" .

Solution

are all examples of a zero matrix XE "zero matrix" .

Tridiagonal matrices XE "Tridiagonal matrices"

XE "matrices" :

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

Example 10

Give an example of a tridiagonal matrix XE "matrix" .

Solution

is a tridiagonal matrix XE "matrix" .

Do non-square matrices XE "matrices" have diagonal XE "diagonal" entries?

Yes, for a matrix XE "matrix" , the diagonal XE "diagonal" entries are where .

Example 11

What are the diagonal XE "diagonal" entries of

Solution

The diagonal XE "diagonal" elements XE "diagonal elements" of are

Diagonally Dominant Matrix XE "Diagonally Dominant Matrix"

XE "Matrix" :

A square matrix XE "square matrix" is a diagonally dominant matrix XE "diagonally dominant matrix" if

for and

for at least one ,

that is, for each row, the absolute value of the diagonal XE "diagonal" element is greater than or equal to the sum of the absolute values of the rest of the elements of that row, and that the inequality is strictly greater than for at least one row. Diagonally dominant matrices XE "matrices" are important in ensuring convergence in iterative schemes of solving simultaneous linear equations.Example 12

Give examples of diagonally dominant matrices XE "matrices" and not diagonally dominant matrices.

Solution

is a diagonally dominant matrix XE "diagonally dominant matrix" as

and for at least one row, that is Rows 1 and 3 in this case, the inequality is a strictly greater than inequality.

is a diagonally dominant matrix XE "diagonally dominant matrix" as

The inequalities are satisfied for all rows and it is satisfied strictly greater than for at least one row (in this case it is Row 3).

is not diagonally dominant as

When are two matrices XE "matrices" considered to be equal?

Two matrices XE "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 for all i and j.

Example 13

What would make

to be equal to

Solution

The two matrices XE "matrices" and ould be equal if and .

04.01.1

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