SQUARE MATRICES. When m = n, (1.1) is square and will be called a square matrix of order n or an n square matrix.
In a square matrix, the elements a11 , a22 , .... , ann are called its diagonal elements.
The sum of the diagonal elements of a square matrix A is called the trace of A.
ZERO MATRIX. A matrix, every element of which is zero, is called a zero matrix. When A is a zero matrix and there can be no confusion as to its order, we shall write A = 0 instead of the m x n array of zero elements.
SUMS OF MATRICES. If A = [aij ] and B = [bij ] are two m x n matrices, their sum (difference), A ± B , is defined as the m x n matrix C = [cij ] , where each element of C is the sum (difference) of the corresponding elements of A and B. Thus, A ± B = [aij ± bij ].
THE IDENTITY MATRIX. A square matrix A whose elements aij = 0 for i > j is called upper triangular ; a square matrix A whose elements aij = 0 for i < j is called lower triangular. Thus
SPECIAL SQUARE MATRICES. If A and B are square matrices such that AB = BA , then A and B are called commutative or are said to commute. It is a simple matter to show that if A is any n-square matrix, it commutes with itself and also with In.
THE INVERSE OF A MATRIX. If A and B are square matrices such that AB = BA = I, then B is called the inverse of A and we write B = A-1 (B equals A inverse). The matrix B also has A as its inverse and we may write A = B-1 .
THE TRANSPOSE OF A MATRIX. The matrix of order n x m obtained by interchanging the rows and columns of an m x n matrix A is called the transpose of A and is denoted by A¢ (A transpose).
SYMMETRIC MATRICES. A square matrix A such that A¢ = A is called symmetric. Thus, a square matrix A = [aij ] is symmetric provided aij = aji , for all values of i and j.
A square matrix A such that A¢ = -A is called skew-symmetric.
HERMITIAN MATRICES. A square matrix A = [aij ] such that Ā¢ = A is called Hermitian. Thus, A is Hermitian provided aij = äij for all values of i and j . Clearly, the diagonal elements of an Hermitian matrix are real numbers.
THE RANK OF A MATRIX. A non-zero matrix A is said to have rank r if at least one of its r-square minors is different from zero while every (r + 1)-square minor, if any, is zero. A zero matrix is said to have rank 0.
EQUIVALENT MATRICES. Two matrices A and B are called equivalent, A~B, if one can be obtained from the other by a sequence of elementary transformations.
ROW EQUIVALENCE. If a matrix A is reduced to B by the use of elementary row transformations a lone, B is said to be row equivalent to A and conversely. The matrices A and B of Example 3 are row equivalent.
THE NORMAL FORM OF A MATRIX. By means of elementary transformations any matrix A of rank r > 0 can be reduced to one of the forms
ELEMENTARY MATRICES. The matrix which results when an elementary row (column) transformation id applied to the identity matrix In is called an elementary row (column) matrix. Here, an elementary matrix will be denoted by the symbol introduced to denote the elementary transformation which produces the matrix.
