21 Introduction to Matrices and Matrix Algebra
Sachin Kumar
Objectives:
1. To familiarise the students with matrices and type of matrices.
2. To let the students carry out the basic operations of matrix algebra.
Introduction:
Matrix or Matrices (in plural) are used in many fields like graph theory, computer science, physics and many more fields. For example in graph theory we use adjacency matrix for finite graph. Each integer value in adjacency matrix represents the number of connection of particular node. Rotation transformation can be represented in matrix form. Matrices are used to solve linear system of equations. Matrices are used in cryptography. In cryptography large matrix is used to encode a message which code is difficult to break. Receiver of encoded message used inverse of matrix to decode the message. Earlier, in optics matrix algebra was used to record reflection and for refraction. Matrices are also used in area of probability and statistics. For example, probability vector which is also a matrix shows the probabilities of diverse outcomes of one experiment. So matrices are important in many fields.
Matrices are also used in environmental sciences. For example, in environmental impact assessment, method, the Leopold matrix is used as tool. It measures the possible impact of a scheme on the environment. The component interaction matrix is used to explore a estuarine system. Similarly there are many other use of matrices in the field of environmental sciences.
1.1 Matrices
Definition: A matrix is a rectangular arrangement/array of objects or elements. The horizontal arrays of a matrix are called its rows and the vertical arrays are called its columns.
is ?×? (to be read as ‘m by n’) matrix with ? rows and ? columns. In compact form the above matrix is represented by ?=[???]?×? or (???)?×?.
We will consider the elements as being real numbers and indicate an element by its row and column position.
The???′? are elements of the matrix ?, where ? indicates the row and ? indicates the column of the element. ?×? is its dimension, where ? is number of rows and ? is number of columns of the matrix.
Example 1.
1.2 Special types of matrices:
1.2.1 Null or zero matrix
If all elements of a matrix are zero then matrix is called null or zero matrix. It is denoted by ??×?.
For example:
1.2.2 Square matrix
A matrix is called square matrix if the number of rows of matrix are equal to the number of columns. A square matrix is said to have order if it has × dimensions.
- In square matrix ?=[???] of nth order,
5
1.2.3 Diagonal matrix
A square matrix is ?=[???] called a diagonal matrix if elements ???=0 for which ?≠?, thus all the elements, except those in the principal diagonal, are zero.
For example:
Example 3.
1.2.4 Scalar matrix
If in diagonal matrix all diagonal entries are same then it is called scalar matrix.
Example 4.
Example 5.
Similarly matrix B of dimension ?×1, is termed as a column vector or column matrix,
Notice, that for addition and subtraction between two matrices A and B, the dimensions of the matrices A and B have to fulfill the following conditions:
number of rows of A= number of rows of B,
number of columns of A=number of columns of B,
Otherwise, addition is not defined.
1.3.3 Scalar Multiplication:
Let ?∈?, then ??=[????]?×?, so all the elements of matrix ? will be multiplied by scalar ?.
Example 8.
Similarly the other parts can also be proved as all the results follow from the properties of real numbers.
Additive Identity and additive Inverse:
Additive Identity: The additive identity of set of matrices of order × is matrix, which, when added to any element in the set, yields.
Additive Inverse: The additive inverse of a matrix of order × is the matrix of order × that, when added to , yields additive identity of set of matrices of order ×.
Let ? be ?×? matrix then
a) For null matrix ??×?, we have 0+?=?=?+0, so here the matrix ??×? is additive identity.
b) There also exist a matrix ? such that ?+?=?=?+?. This matrix ? is called additive inverse of matrix ? and it is denoted as −?
1.3.4 Multiplication:
Two matrices A and B can be multiplied, only if the number of columns of A = number of rows of B. If this condition is fulfilled, then it is possible to define the product AB.
??? element of product matrix (??)= sum of the product of elements of ??ℎ row of ? with the corresponding elements of ??ℎ column of ?.
NOTE If ? and ? are two matrices such that ?? exists, the ?? may or may not exist.
NOTE In this case ?? does not exist, because the number of column in ? is not equal to the number of rows in ?.
Theorem 1. Matrix multiplication is not commutative in general.
i.e. ??≠?? as shown in above Example 9, ?? exist whereas ?? does not exist.
Theorem 2. Matrix multiplication is associative i.e. (??)?=?(??), whenever both sides are defined.
Theorem 3. Matrix multiplication is distributive over matrix addition i.e.,
(i) ?(?+?)=??+??
(ii) (?+?)?=??+?? whenever both sides of equality are defined.
Theorem 4. If ? is an ?×? matrix, then ???=?=???.
Properties of transpose:
1. For any matrix ?,(??)?=?.
2. For any two matrices ? and ? of the same order, (?+?)?=??+??.
3. If ? is a matrix and ? is a scalar, then (??)?=?(?)?.
4. If ? and ? are two matrices such that ?? is defined, then (??)?=????
1.3.6 Symmetric and Skew-Symmetric Matrices
1.3.6.1 SYMMETRIC MATRIX A square matrix ?=(???) is called a symmetric matrix, if ???=??? for all ?,?.
It follows from definition of a symmetric matrix that ? is symmetric
If and only if ???=??? for all ?,?
If and only if ?=??.
Thus, a square matrix ? is symmetric matrix if and only if ??=?.
Example 16. Every square matrix can be uniquely expressed as the sum of a symmetric and a skew-symmetric matrix.
SOLUTION Let ? be a square matrix. Then,
Thus ?=?+?, where ? is a symmetric matrix and ? is a skew-symmetric matrix. Hence ? is expressible as the sum of symmetric and skew-symmetric matrix.
Thus ? can be expressed as sum of symmetric and skew-symmetric matrix.
Example: A matrix which is both symmetric as well as skew symmetric is null matrix.
Summary:
In this module, we have introduces the concept of matrices. We have discussed the various types of matrices like null matrix, square matrix, diagonal matrix, scalar matrix, unit matrix, triangle matrix, diagonally dominant matrix. We have also discussed the some basic laws of matrix algebra. We have discussed that when two matrices are equal. Also we have discussed when two matrices can be added or subtracted or multiplied. Then we have introduced the concept of transpose of a matrix and its properties. It is also shown that when will a matrix be symmetric or skew-symmetric. Then it is shown that every square matrix can be written as sum of symmetric and skew symmetric matrix.
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Suggested Books for reading:
[1] Kreyszig, Erwin. Advanced engineering mathematics. John Wiley & Sons, 2010.
[2] Lang, Serge. Linear Algebra. Springer-Verlag, 1987.
[3] Bronson, Richard. Matrix Methods: An Introduction. Academic Press,