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

By C. G. Broyden

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Since this orthogonality is independent of the ordering of the rows the lemma is established. 28 BASIC MATRICES 2. 2 The Unit Matrix In the field of real numbers we know that there is a number, one, that leaves any other number unchanged when multiplied by it. This number is sometimes referred to as unity, or the unit element. We now consider the possibility of the existence of a matrix that will leave any matrix unchanged when pre-multiplied by it, and examine its properties. 2 A unit lrUltrix I is a matrix such that the matrix product IA is equal to A for any matrix A for which the product is defined.

9 A spanning set consisting of linearly independent vectors is called a basis. It follows then that a basis consists of any n linearly independent vectors. 6, and more than n are necessarily linearly dependent and so fail to satisfy the definition. 7, if a set of vectors forms a spanning set when it does not include n linearly independent vectors. The answer is in the negative but further discussion of this case is deferred until chapter 7. 7 The Sherman-Morrison Formula Many computational and theoretical procedures rely on the modification of a matrix by adding a term of the form xy T, where x and yare vectors of suitable order.

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. This is accomplished as follows. 3) and since II x II =F 0 the result follows. 4).

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