Mathematical Methods for Physics and Engineering - Matematica.NET

MATRICES AND VECTOR SPACES◮The matrices A, B and C are given by( ) ( ) ( )2 −11 0−2 1A =, B =, C =.3 10 −2−1 1Find the matrix D = A +2B − C.(2 −1D =3 1=) (1 0+20 −2) (−2 1−−1 1(2+2× 1 − (−2) −1+2× 0 − 13+2× 0 − (−1) 1+2× (−2) − 1))=( )6 −2. ◭4 −4From the above considerations we see that the set of all, in general complex,M × N matrices (with fixed M and N) forms a linear vector space of dimensionMN. One basis for the space is the set of M × N matrices E (p,q) with the propertythat E (p,q)ij =1ifi = p and j = q whilst E (p,q)ij = 0 for all other values of i andj, i.e. each matrix has only one non-zero entry, which equals unity. Here the pair(p, q) is simply a label that picks out a particular one of the matrices E (p,q) ,thetotal number of which is MN.8.4.2 Multiplication of matricesLet us consider again the ‘transformation’ of one vector into another, y = A x,which, from (8.24), may be described in terms of components with respect to aparticular basis asy i =N∑A ij x j for i =1, 2,...,M. (8.32)j=1Writing this in matrix form as y = Ax we have⎛ ⎞⎛⎛⎞y 1 A 11 A 12 ... A 1Ny 2⎜ ⎟⎝ . ⎠ = A 21 A 22 ... A 2N⎜⎝.. ⎟. .. . ⎠ ⎜⎝A M1 A M2 ... A MNy Mx 1x 2..x N⎞⎟⎠(8.33)where we have highlighted with boxes the components used to calculate theelement y 2 : using (8.32) for i =2,y 2 = A 21 x 1 + A 22 x 2 + ···+ A 2N x N .All the other components y i are calculated similarly.If instead we operate with A on a basis vector e j having all components zero252