By C. G. Broyden
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Extra resources for Basic Matrices: An Introduction to Matrix Theory and Practice
To show that the Euclidean norm satisfies the triangular inequality is a little more difficult, and we proceed by first proving another important inequality. Cauchy's Inequality For any two non-nui! ;;; II x Ib II y 112 with equality if and only if y is a scalar multiple of x. MATRIX AND VECTOR NORMS Proof 43 Let Then M is symmetric and M2 = M. Let z = My. Then yTMy = yTM2y = zT z;;;' 0, so that YTMy = YTY - (y T X)2 -T- ;;;. 0 X X with equality if and only if z is null. It follows immediately from the above inequality, since xTx> 0, that (x Tx)(yT y);;;'(yT x)2 or Ilx11211y112;;;'l yTx l with equality if and only if z is null.
At best, since k(A) ~ 1, a comparable relative error to that of b is always possible. 11 only gives, of course, an upper bound for the error and often this upper bound is not even remotely approached, although it is possible to find values of band Sb for which the bound is attained. 11 though is that it is unrealistic to expect a higher accuracy in the solution of a set of linear simultaneous equations than exists in the data, and sometimes one must be content with a substantially lower one. In many practical cases the elements of the vector b have been obtained by measurement, graph-reading or numerical calculation, and so will be inherently inaccurate.
2 (The Banach Lemma) Let B satisfy the conditions of the previous lemma. Then I II (I + B)-III ,,;;;--I -- II B II Proof Taking norms and applying the triangular inequalities to the identity (I+B)-I =1-8(1+B)-1 MATRIX AND VECTOR NORMS 49 gives II (I + 8)-1 II";;; I + II 81111 (I + 8)-1 II from which, since II 8 II < 1, the lemma is immediately deduced. 6 Errors in the Solu tion of Linear Equations We now investigate how the solution of the set of linear equations Ax = b, where A is nonsingular, is affected by perturbations in band A.