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

By C. G. Broyden

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2) where, for our purposes, p = I, 2 or 00. p/II x il p' x 0 does in fact possess a maximum value that is attained for a particular choice of x. 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.

7 Show that if A, B, C and D are matrices such that 0 then = ABC, and 0 = [d ij ] d ij = alBcj where aF is the ith row of A and Cj is the jth column of C. 8 Prove that, if A is real, AT A is symmetric and if A is complex, AHA is Hermitian. 9 Let A = [aij] be an 11th order matrix. Then the trace of A, written tr(A), is the sum of its diagonal elements. Show that (a) tr(A + B) = tr(A) + tr(B); (b) if x and yare 11th order vectors then tr(xy T) = XTy. 1 0 Let A and B be any two matrices for which the products AB and BA are both defined.

Proof The corresponding matrix A has more rows than columns, hence the former are linearly dependent from the fundamental theorem. We now prove that n or more vectors of order n do form a spanning set provided that certain conditions are satisfied. Theorem 2. 7 A set of n or more vectors of order n is a spanning set if it includes n linearly independent vectors. Proof Let the n linearly independent vectors form the columns of the matrix Al and the remainder (if any) form the columns of A 2 . 7.

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