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CHAPTER V. MATRICES
- Hong Kong University Press, HKU
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CHAPTER V MATRICES In Examples 5.8, we gave some effective methods of constructing linear transformations. Among others, we saw that for any fmitedimensional linear spaces X and Y over A with bases (X,, ..., xm) and (y,, . .., yn) respectively a unique linear transformation @: X - + Y is determined by a family ( ~ ~ j ) ~ . , , ..,m , . j =l , .. . , n of scalars in such a way that 4(Xi)=UilY1 + ...+Uinyn for i = l , ...,m. Conversely, let $:X +Y be an arbitrary linear transformation. In writing each $ (xi) as a linear combination of the base vectors yj of Y, i.e., we obtain a family 1 , . ..,m;j= ,,..., n of scalars. Thus relative to the bases (X,, . . . , xm)and (y, ,... , y, )of X and Y respectively, each linear transformation @: X - + Y is uniquely characterized by a family (aij)i=l , . .., , ..., , of mn scalars of A. This therefore suggests the notion of a matrix as a doubly indexed family of scalars. Matrices are one of the most important tools in the study of linear transformations on finite-dimensional linear spaces. However, we need not overestimate their importance in the theory of linear algebra since the matrices play for the linear transformations only a role that is analogous to that played by the coordinates for the vectors. 813. General Properties of Matrices A. Notations DEFINITION 13.1. Let p and q be two arbitrary positive integers. A real (p, 4)-matrix is a doubly indexed family M =(pij)i=l , .. ., p; j = l , ..., q of real numbers. A complex (p, 4)-matrix is similarly defined. Here again we use 156 V MATRICES the term "M is a matrix over A" to mean that M is a real matrix or a complex matrix according to whether A =R or A =C. It is customary to write a @,q) -matrix M = ( p ~ ~ ~ ) ~ = ~ , ... ,p; j= , . ,, in the form of a rectangular array thus: From the definition, it follows that two matrices and are equal if and only if (i) p =r and q =S .and(ii) pij=Vijfor every pair of indices i and j. We introduce now the following notations and abbreviations to facilitate references and formulations. (a) The ordered pair (p, q) is called the size of the matric M. If this is clear from the context, then we shall also write M =p,,. For practical purposes, we do not distinguish the (1, l)-matrix (X) from the scalar X. (b) The scalar Pij is called a term of the matrix M and the indices i and j are respectively called the row index and column index of the term Pij. . . . . (c) For every row index i = 1, q, the family pi* [3.17.28.48] Project MUSE (2024-04-24 16:48 GMT) 513 GENERAL PROPERTIES OF MATRICES 157 - - @ij)j=~, . . ., q io called the i-th row of M. Clearly pi* is a (l, q)-matrix over A. On the other hand pi* is also a vector of the arithmetical linear space Aq; therefore we may also refer to pi* as the i-th row vector of M. (d) For every column index j = 1, ...,q, the family P*j =(pij)i=~, . . . , p is called the j-th column ofM Clearly p,/ is a (p, 1)-matrix over A.On the other hand p*i is also a vector of the arithmetical linear space A*; therefore we may also refer to p*j as the j-th column vector of M. (e) The term pii is therefore said to belong to the i-th row and thej-th colomn of M. (0Thediagonal ofM isthe orderedp- tuple (pl 1, p22 ,. .., ppp) if p Q q and it is the ordered q-tuple (p, ,,p,, ,... , pqp) if p 2 q. EXAMPLE 13.2. Consider a rotation in the ordinary plane about the origin 0 by an angle 8. The point P with cartesian coordinates(X,y ) istaken into the point P' with cartesian coordinates (X',y'), where We call the (2,2)- matrix of rotation 158 V MATRICES EXAMPLE 13.3 Let M = p** be a (p,q)-matrix. Consider the (q,p)-matrix M' =p'** where pij =$ S r for all i = 1, . .., p and j =1, ..., q.The matrix M' is called the transpose of the matrix M and is obtained by turningM around about the diagonal. . ... .. ...... ...... \ \ Ppp \ Between the row vectors and the column vectors we have =p',, and p*j =pfj*.Moreover, for any matrix M, (M')' = M. EXAMPLE13.4. Let...
