Introduction to Matrices

1
Introduction to Matrices
2
What you need to know

• General algebraic methods
• Use of functions
• Some knowledge of vectors and how to manipulate
  them
• IMPORTANT NOTE MAKE SURE YOU ARE COMFORTABLE
  WITH THE ABOVE BEFORE PROCEEDING WITH THIS
  SECTION.

3
MATRICES A TEXTBOOK DEFINITION

• A matrix is a set of quantities (called elements
  or entries) arranged in a rectangular array, with
  certain rules governing their combination.
  Conventionally, the array is enclosed in round
  brackets or, less commonly, in square brackets.
• (Penguin Dictionary of Mathematics, Second
  Edition, Ed. David Nelson, Penguin Books Ltd
  1998, p 271)
• ALL WILL BECOME CLEAR AS THIS UNIT CONTINUES!

4
The Elements of a Matrix

• As was described by the textbook definition on
  the previous page, a matrix is a rectangular
  array of numbers, and can be a useful way of
  storing information. For this course, matrices
  shall be written in square brackets, although
  they are more commonly written in parentheses.
• This is an example of a matrix, which shall be
  referred to as A
• 1 3 5 7 A 2 4 6
  8 5 9 0 2
• We would call this a 3x4 Matrix, because it has 3
  rows and 4 columns. In the same way, an i x j
  matrix is one with i rows and j columns.

5
The Elements of a Matrix 2

• By convention, matrices are named with an upper
  case letter, e.g. A, and are often shown
  emboldened.
• Consider A again A 1 3 5 7 2 4
  6 8 5 9 0 2
• Each separate part of a matrix is called an
  element.
• The element in the ith row and jth column of A is
  referred to as ai j. For example, the element in
  the second row and third column, 6, is a2 3.

6
The Information in a Matrix

• Matrices can be used to store all sorts of
  information.
• For example, consider we run a small hardware
  store, and have three different products that we
  want to observe the sales of, over the course of
  four days. A can now be considered as a form of
  table, in which we have stored the information
  Mon Tues Wed Thurs PANS 1 3
  5 7 NAILS 2 4 6
  8 SAWS 5 9 0 2

7
The Information in a Matrix 2

• As you have seen matrices can be used to store
  various kinds of information. Mathematical
  operations can also be performed with them.
• We shall return to this hardware example in the
  second section of this course, where we shall
  consider what we can do with the information
  contained within a matrix.
• For the rest of this section we shall expand our
  knowledge of matrix terminology and learn about
  some important types of matrix.

8
Special Types of Matrix Same Kind

• We can say that two matrices are of the same
  kind, when they are both of the kind n x m, i.e.
  they have the same number of rows and columns.
• For example, A and B, shown below, are of the
  same kind because A has as many rows as B and A
  has as many columns as B. A 1 3
  5 7 B 5 2 4 7 2 4
  6 8 6 4 8 9 5 9 0 2 0 9
  0 7
• Matrices are therefore of the same kind when they
  have the same structure with the same number of
  elements.

9
Special Types of Matrix

• The rest of this introductory unit will provide
  you with some of the different types of special,
  or named, matrix.
• This is in order to prepare you for mathematical
  operations on matrices, which will be covered in
  the next unit, and will involve the use of the
  following special matrices.
• Try to learn the different types of matrix in
  preparation.

10
Special Types of Matrix - Vector

• There are two types of vector matrix, the Column
  Vector and the Row Vector.
• A column vector is a matrix which only has one
  column (the number of rows is not taken into
  consideration).
• For example C 1 is an example of a
  3x1column matrix, 2 or column
  vector. 5

11
Special Types of Matrix Vector 2

• In contrast, a row vector is a matrix with only
  one row.
• An example of a 1x8 row matrix, or row vector,
  is given below D 1 3 5 7 2 4 6
  8
• So, we have shown you matrices with either a
  single row or a single column, but what about a
  1x1 matrix, which stores only one unit of
  information? In this instance, it is worth
  bearing in mind that there is no difference
  between this matrix and an ordinary number.

12
Special Types of Matrix - Square

• A matrix is called a Square Matrix if we can say
  that it has n rows and n columns. This means
  that it has an equal number of rows and columns.
  Examples of two square matrices are given
  below E 1 3 5 F 5 2
  4 7 2 4 6 6 4 8 9 5
  9 0 0 9 0 7 1 5 0 8
• Here E is a 3x3 matrix and F is a 4x4 matrix.
• In a square matrix the elements leading
  diagonally down from the top left corner are
  called diagonal elements, e.g. the elements at
  points 1,1 2,2 3,3 etc.

13
Special Types of Matrix Zero

• When all the elements of a matrix are 0, we call
  the matrix a 0-matrix, or zero-matrix. For
  example G 0 0 0 H 0
  0 0 0 0 0 0 0 0 0
  
• In the example above, G is a square zero-matrix
  and H is a row-vector zero matrix.

14
Special Types of Matrix - Diagonal

• We have already seen that square matrices have a
  leading diagonal. This is an important feature
  in matrices, and particular attention should be
  paid to them.
• A Diagonal Matrix is a square matrix in which all
  the non-diagonal elements are 0. Such matrices
  are completely denoted by the diagonal elements.
  An example is I 7 0 0 0
  4 0 0 0 9
• In the example above the matrix is defined by the
  leading diagonal, and is thereby denoted by
  diag(7 , 4 , 9)

15
Special Types of Matrix - Diagonal 2

• There are different types of diagonal matrix
• A Scalar Matrix is a diagonal matrix in which all
  the diagonal elements are the same. For example
  J 5 0 0 0 5 0
  0 0 5
• An Identity Matrix is a special type of scalar
  matrix in which the leading diagonal is entirely
  composed of 1s. For example K 1 0 0
  0 1 0 0 0 1
• Scalar Matrices, and especially Identity
  Matrices, are very important in matrix
  manipulation, and will be covered in the next
  section.

16
Other Special Matrices

• Given any matrix, there are transformations which
  we can perform on it, and these shall be
  described next.
• These transformations count as special types of
  matrices in their own right.
• We shall consider the transpose, opposite and
  skew-symmetric matrices of a matrix.

17
The Transpose of a Matrix

• The matrix A is the transpose of the matrix A
  and the matrix F is the transpose of the matrix
  F
• A 1 3 5 7 F 5 2 4 7 2
  4 6 8 6 4 8 9 5 9 0 2 0 9
  0 7 1 5 0 8
• A 1 2 5 F 5 6 0 1 3
  4 9 2 4 9 5 5 6 0 4 8 0
  0 7 8 2 7 9 7 8
• Can you see what has been done here?

18
The Transpose of a Matrix 2

• Hopefully you could see what had happened there.
• The transpose of a n x m matrix A is the matrix
  A when the ith row of A is the ith column of A'
  for the entire matrix. So the elements which in
  matrix A were at (i,j) are at (j,i) for matrix
  A.
• To denote the transpose of A we can write T(A) or
  AT.

19
The Transpose of a Matrix 2

• It is hoped that you also noticed that for the
  square matrix F, the elements on the leading
  diagonal remained unchanged. This is an
  important result and it leads to the concept of
  symmetric matrices.
• A square matrix can be called symmetric if it is
  equal, or identical, to its transpose. All
  Identity matrices are symmetric. For example L
  4 3 K 1 0 0 3 4 0
  1 0 0 0 1 LT 4 3
  KT 1 0 0 3 4 0 1 0
  0 0 1

20
The Opposite Matrix of a Matrix

• The opposite matrix of a matrix is when we change
  the sign of all the elements of the matrix
• A 1 3 5 7 M 5 -2 4 0 2
  4 6 8 6 2 -8 9 5 9 0 2 0 -9
  0 7 1 6 0 -3
• -A -1 -3 -5 -7 -M -5 2 -4
  0 -2 -4 -6 -8 -6 -2 8 -9 -5 -9 0
  -2 0 9 0 -7 -1 -6 0 3
• If there are any 0s in the matrix then they
  should be left as they are.

21
A Skew-Symmetric Matrix

• A square matrix is called skew-symmetric if it is
  equal, or identical, to the opposite of its
  transpose.
• For example O 0 -5 -1 -7 OT
  0 5 1 7 -OT 0 -5 -1 -7 5 0 -5
  -1 -5 0 5 1 5 0 -5 -1 1 5 0
  -5 -1 -5 0 5 1 5 0 -5 7 1 5
  0 -7 -1 -5 0 7 1 5 0
• Can you see how this works? Think about it.

22
Exercise 1 See Answers.ppt

• Work through the slideshow until you are
  confident that you understand the different types
  of matrices and how they work. This is very
  important for the next section.
• Identify the different types of matrix shown on
  the next page.
• Label each one with size details and name, e.g.
  4x5 matrix, or 6x6 identity matrix.
• Find the Transpose and the Opposite of matrices
  a, b and c.

23
Exercise 1 Continued

• 5 6 7
• 5 6 7
• 10 12 15 17 7
  15
• 9 8 4 5
• 7 0 0 0 5 0
  0 0 2

24
Exercise 1 Continued

• 0 0 0 0
• 5 6 2 4 1 5 7 3
  7 1 5 2 1 2 8 5
• 1 0 0 0 1 0
  0 0 1
• 5 0 0 0 5 0
  0 0 5

25
Summary

• In this section you have learnt how to recognise
  different types of matrices.
• In the next section you will see how matrices can
  be used to store information, and how to perform
  simple mathematical calculations with them.
  Special matrices will be used to illustrate
  certain points, and more practical exercises will
  be given.
• When you are satisfied that you have understood
  this section go to Mathematical Operations on
  Matrices.