By C. G. Broyden
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We shall find in fact that if this number is to be of any use to us at all it must satisfy a third condition - the triangle inequality - so called because in one form it states that the length of 42 BASIC MATRICES one side of a triangle cannot exceed the combined lengths of the other two sides. This number that we would associate with a vector we call a norm. ;;; II x II + II y II (triangle inequality) We shall be concerned here with three different vector norms. 2 If x = [x;l the 1\,1 2 and I~ norms of x are defined by (a) II x 11\ = ~ I Xi I, the 1\ i (b) II x 112 = (f I xi I) norm \/2 ,the 12 or Euclidean norm (c) II x II~ = m~x I xi I, the I I~, infinity or uniform norm Of these the second is the one that is most commonly used.
II xiiI il Y III is needlessly pessimistic. 9). To avoid error when using the II or I~ norms it is helpful to regard them as 'columnsum' or 'rowsum' norms respectively. 7 rather than by the formal definitions, and are consistent in our usage, then no blunders need occur. Before discussing some results obtained from the usc of norms we define one more matrix norm, the Frobenius norm, which we distinguish by the subscript F. / It is thus the matrix analogue of the Euclidean vector norm but flut, however, the norm induced by that vector norm.
The principal effect of a large condition number therefore is to increase the uncertainty of the relative error of the solution for a given relative error in the data. The possibility, however, that large errors may occur cannot be too highly stressed, and should constantly be borne in mind by all those engaged in practical computation and those who use any results so obtained. 50 BASIC MATRICES Another application of this result arises when testing the accuracy of some approximate solution of the set of linear equations Ax = b.