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Chapter One
MATRIX ALGEBRA
In this first chapter we shall introduce one of the principal objects of study in linear algebra, a matrix or rectangular array of numbers, together with the standard matrix operations. Matrices are encountered frequently in many areas of mathematics, engineering, and the physical and social sciences, typically when data is given in tabular form. But perhaps the most familiar situation in which matrices arise is in the solution of systems of linear equations.
1.1 Matrices
An m × matrix A is a rectangular array of numbers, real or complex, with m rows and n columns. We shall write ciij for thenumber that appears in the ith row and the jth column of A; this is called the (i,j) entry of A. We can either write A in the extended form
or in the more compact form
Thus in the compact form a formula for the (i,j) entry of A is given inside the round brackets, while the subscripts m and n tell us the respective numbers of rows and columns of A.
Explicit examples of matrices are
Example 1.1.1
Write down the extended form of the matrix
.
The (i, j) entry of the matrix is (—l)lj + i where i — 1, 2, 3, and j — 1,2. So the matrix is
It is necessary to decide when two matrices A and B are to be regarded as equal; in symbols A = B. Let us agree this will mean that the matrices A and B have the same numbers of rows and columns, and that, for all i and j, the (i,j) entry of A equals the (i,j) entry of B. In short, two matrices are equal if they look exactly alike.
As has already been mentioned, matrices arise when one has to deal with linear equations. We shall now explain how this comes about. Suppose we have a set of
m linear equations in
n unknowns
. These may be written in the form
Here the a
ij and are to be regarded as given numbers. The problem is to
solve the system, that is, to find all
n-tuples of numbers
that satisfy every equation of the system, or to show that no such numbers exist. Solving a set of linear equations is in many ways the most basic problem of linear algebra.
The reader will probably have noticed that there is a matrix involved in the above linear system, namely the coefficient matrix
In fact there is a second matrix present; it is obtained by using the numbers bi, b2,…, bm to add a new column, the (n + l)th, to the coefficient matrix A. This results in an m x (n + 1) matrix called the augmented matnx of the linear system. The problem of solving linear systems will be taken up in earnest in Chapter Two, where it will emerge that the coefficient and augmented matrices play a critical role. At this point we merely wish to point out that here is a natural problem in which matrices are involved in an essential way.
Example 1.1.2
The coefficient and augmented matrices of the pair of linear equations
are respectively
Some special matrices
Certain special types of matrices that occur frequently will now be recorded.
(i) A 1 × n matrix, or — row vector, A has a single row
(ii) An m × 1 matrix, or m-column vector, B has just one column
(iii) A matrix with the same number of rows and columns is said to be square.
(iv) A zero matrix is a matrix all of whose entries are zero. The zero m × n matrix is denoted by
Sometimes 0nn is written 0n. For example, O23 is the matrix
(v) The identity nxn matrix has l's on the principal diagonal, that is, from top left to bottom right, and zeros elsewhere; thus it has the form
This matrix is written
The identity matrix plays the role of the number 1 in matrix multiplication.
(vi) A square matrix is called upper triangular if it has only zero entries below the principal diagonal. Similarly a matrix
is lower triangular if all entries above the principal diagonal are zero. For example, the matrices
are upper triangular and lower triangular respectively.
(vii) A square matrix in which all the non-zero elements lie on the principal diagonal is called a diagonal matrix. A scalar matrix is a diagonal matrix in which the elements on the principal diagonal are all equal. For example, the matrices
are respectively diagonal and scalar. Diagonal matrices have much simple...
