By C. G. Broyden
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Additional info for Basic Matrices: An Introduction to Matrix Theory and Practice
2 that if the columns of A are linearly dependent the vector of multipliers x is not unique. Post·multiplication of that equation by the scalar e gives AxO = 0 so that x may always be scaled by any arbitrary non-zero quantity without affecting its orthogonality to the rows of A. 3 below). We now establish a fairly trivial lemma which tells us that we can permute the rows of A without affecting the linear dependence or otherwise of the columns. 1 The linear independence or otherwise of the columns of A is unaltered if the rows of A are permuted.
5 Let A be a square matrix, written in partitioned form A= [All A2 I A12] A22 Show that (a) if Al I is square and nonsingular then A is nonsingular if and only if(A2 2 - A21 All A 12 ) is nonsingular; (b) if AI2 is square and nonsingular then A is nonsingular if and only if (A2 I - A2 2 A ~1 Al I) is Singular. If Al I is square and nonsingular, A -I = Band B is partitioned identically to A, obtain the partitions Bi; in terms of the partitions Ai;. Check your result by verifying the special cases (a) A2 I = 0 (b) A2 I = 0 and Al 2 = 0 (c) A2 2 = o.
4 Prove that, for any subordinate norm, II AB II ,;;;;; II A II II B II. 5 Let A = [aiil. Show that max I aii I is not a norm of A. i. 6 Show that, for the 11 ,1 2 and L norms, the norm of a matrix cannot be less than the norm of any of its submatrices. 7 Show that II A - B II;;' III A II -II Bill. 8 Show that, if x is an nth order vector, II x II" ,;;;; II x III ,;;;; n II x II~ and II x II~ ,;;;; II x 112 ,;;;; n l/2 11 x II~. Hence infer that if, for the sequence of vectors ei ,lim II ei II = 0 for anyone of the above three norms then lim II ei II = 0 for i+oo i~(X) the remaining two norms.
Basic Matrices: An Introduction to Matrix Theory and Practice by C. G. Broyden