Advanced Calculus fi..

18 Advanced Calculus, Fifth EditionBy a matrix we mean a rectangular array of m rows and n columns:For this chapter (with a very few exceptions) the objects al a12, . . . , a,,will bereal numbers. In some applications they are complex numbers, and in some they arefunctions of one or more variables. We call each a;, an entry of the matrix; morespecifically, ai, is the i j-entry.We can denote a matrix by a single letter such as A, B, C, X, Y, .... If A denotesthe matrix (1.48), then we write also, concisely, A = (a;,).Let A be the matrix (aij) of (1.48). We say that A is an m x n matrix. Whenm = n, we say that A is a square matrix of order n. The following are examples ofmatrices:Here A is 2 x 3, and B and C are 2 x 2; B and C are square matrices of order 2.An important square matrix is the identity matrix of order n, denoted by I:0 0 ...1 for i = j,0 fori# j.(1.50)We call 6;; the Kronecker delta symbol. One sometimes writes I, to indicate theorder of I, but normally the context makes this unnecessary.The principal diagonal of a square matrix is formed of the entries all,a22, . . . , a,,. For I, the principal diagonal is 1, 1,. . . , 1 (n times).For each m and n we define the m x n zero matrixOne sometimes denotes the matrix by Om, to indicate the size-that is, the numberof rows and columns.In general, two matrices A = (ai,) and B = (bij) are said to be equal, A = B,when A and B have the same size and aij = b;; for all i and j.A 1 x n matrix A is formed of one row: A = (all,.. ., a,,). We call sucha matrix a row vector. In a general m x n matrix (1.48), each of the successiverows forms a row vector. We often denote a row vector by a boldface symbol: u,v,. ..(or, in handwriting, by an arrow). Thus the matrix A in (1.49) has the rowvectors ul = (2, 3,5) andu2 = (1,2,3).