Mathematical Methods for Physics and Engineering - Matematica.NET

8.3 MATRICES8.2.1 Properties of linear operatorsIf x is a vector and A and B are two linear operators then it follows that(A + B )x = A x + B x,(λA )x = λ(A x),(AB)x = A (B x),where in the last equality we see that the action of two linear operators insuccession is associative. The product of two linear operators is not in generalcommutative, however, so that in general ABx ≠ BAx. In an obvious way wedefine the null (or zero) and identity operators byO x = 0 and I x = x,for any vector x in our vector space. Two operators A and B are equal ifA x = B x for all vectors x. Finally, if there exists an operator A −1 such thatAA −1 = A −1 A = Ithen A −1 is the inverse of A . Some linear operators do not possess an inverseand are called singular, whilst those operators that do have an inverse are termednon-singular.8.3 MatricesWe have seen that in a particular basis e i both vectors and linear operatorscan be described in terms of their components with respect to the basis. Thesecomponents may be displayed as an array of numbers called a matrix. Ingeneral,if a linear operator A transforms vectors from an N-dimensional vector space,for which we choose a basis e j , j =1, 2,...,N, into vectors belonging to anM-dimensional vector space, with basis f i , i =1, 2,...,M, then we may representthe operator A by the matrix⎛⎞A 11 A 12 ... A 1NA 21 A 22 ... A 2NA = ⎜⎝. ⎟. . .. . ⎠ . (8.25)A M1 A M2 ... A MNThe matrix elements A ij are the components of the linear operator with respectto the bases e j and f i ; the component A ij of the linear operator appears in theith row and jth column of the matrix. The array has M rows and N columnsand is thus called an M × N matrix. If the dimensions of the two vector spacesare the same, i.e. M = N (for example, if they are the same vector space) then wemay represent A by an N × N or square matrix of order N. The component A ij ,which in general may be complex, is also denoted by (A) ij .249