Basic Matrices: An Introduction to Matrix Theory and by C. G. Broyden

By C. G. Broyden

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We shall prove later (much later in the case of the 12 norm) that this is indeed the case for the I. ,1 2 and I~ norms. 3) (from now on we write II . II for II . lip) with II Ax 1/ = II A III/x II for at least one value of x. We shall use this last result to establish the triangle inequality for subordinate matrix norms. 2 then II A II = 0 only if A is null, and II A II > 0 otherwise. h is also clear that II A II satisfies the homogeneity condition so it remains to show that the triangle inequalities are satisfied.

Then it is readily verified that so that A is nonsingular; (b) Let A I I be singular. Then there exists a vector x '1= 0 such that 34 BASIC MATRICES All X = O. Hence and A is singular. If A2 2 is singular the singularity of A follows from the existence of a vector y 1= 0 such that y T A2 2 = OT, completing the proof. 6 An nth order upper triangular matrix V = [Uij] is nonsingular if and only if uii1=O,I~i~n. Proof Let Vi denote the ith order leading principal submatrix of V, so that V= Vn and Vi = [ViOl v· ] U'ii ' 2 ~i~n where vT = [Uli' u2j, ...

N. The reason for the use of terms like 'upper triangular' will become obvious if one or two actual examples are written down. Their principal features are the ease with which linear equations involving them may be solved, and the fact that their singularity or otherwise may be determined by inspection. In order to investigate the second of these phenomena we first prove the following lemma. 4 Let A be a square matrix, where and where the submatrices Al I and A2 2 are themselves square. Then A is nonsingular if and only if both All and A2 2 are nonsingular.

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