Mathematical Methods for Physics and Engineering - Matematica.NET

8.4 BASIC MATRIX ALGEBRANow, since x is arbitrary, we can immediately deduce the way in which matricesare added or multiplied, i.e.(A + B) ij = A ij + B ij , (8.26)(λA) ij = λA ij , (8.27)(AB) ij = ∑ A ik B kj .k(8.28)We note that a matrix element may, in general, be complex. We now discussmatrix addition and multiplication in more detail.8.4.1 Matrix addition and multiplication by a scalarFrom (8.26) we see that the sum of two matrices, S = A + B, is the matrix whoseelements are given byS ij = A ij + B ijfor every pair of subscripts i, j, with i =1, 2,...,M and j =1, 2,...,N. Forexample, if A and B are 2 × 3 matrices then S = A + B is given by( ) ( )S11 S 12 S 13 A11 A=12 A 13+S 21 S 22 S 23 A 21 A 22 A 23=( )B11 B 12 B 13B 21 B 22 B 23( )A11 + B 11 A 12 + B 12 A 13 + B 13. (8.29)A 21 + B 21 A 22 + B 22 A 23 + B 23Clearly, for the sum of two matrices to have any meaning, the matrices must havethe same dimensions, i.e. both be M × N matrices.From definition (8.29) it follows that A + B = B + A and that the sum of anumber of matrices can be written unambiguously without bracketting, i.e. matrixaddition is commutative and associative.The difference of two matrices is defined by direct analogy with addition. Thematrix D = A − B has elementsD ij = A ij − B ij , for i =1, 2,...,M, j =1, 2,...,N. (8.30)From (8.27) the product of a matrix A with a scalar λ is the matrix withelements λA ij , for example( ) ( )A11 Aλ12 A 13 λA11 λA=12 λA 13. (8.31)A 21 A 22 A 23 λA 21 λA 22 λA 23Multiplication by a scalar is distributive and associative.251