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

By C. G. Broyden

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Then (A + xy T)A -1 x = x(l + YT A -1 x) = O. Since x ~ 0 and A-I is nonsingular, A -1 X ~ 0 so that there exists a non-null vector which, when pre-multiplied by (A + xy T), gives the null vector. (A + xy T) is thus singular. (b) Assume yT A -1 X T -1. Then, as may be readily verified (A + xy T) ( A-I - A -1 xy T A-I ) _ l+yTA IX =I Hence A + xy T possesses an inverse and is thus nonsingular, and the theorem is proved. The expression for the inverse of the modified matrix (A + xy T ) -1 = A -1 _ T A-I Y 1 + YT A-I X A -1 X is known as the Sherman-Morrison formula.

If Q2 those values of the scalar 0 for which {P + pOq T)2 = O. = 0 determine 2 Some Elementary Consequences of Linear Independence 2. J Linear Independence The idea of 'linear independence' is fundamental not only to matrix algebra but to other branches of mathematics as well. It is, indeed, one of the really basic notions of mathematics and much of what follows will rely on the consequences of a set of vectors being linearly independent or otherwise. We begin therefore with a formal definition of this key idea.

As often as not the matrix to be modified, A, say, is an n x n nonsingular matrix (so that x and yare both nth order vectors) and although we know the modification to A we are primarily concerned with what happens to the inverse. It is, in fact, convenient to be able to express (A + xy1) -1 in terms of A-I, and this is preCisely what the Sherman-Morrison formula accomplishes. 8 Let A be nonsingular. Then (A + xy T) is singular if and only if y T A -1 X = -1. Proof (a) Assume y TA -1 X = -1. Then (A + xy T)A -1 x = x(l + YT A -1 x) = O.

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