The Approximate Solution of Matrix Problems

A ( x Au) = h Ahu. Hence if X is small, any component of error along u can be completely obscured in the rounding process, so tha t an "approx imate" solution x* =" ~c Au, even if crude, m a y satisfy the system exactly to within machine errors. The situation is the same in the inversion of a matr ix; in fact, the following theorem is of interest: THEOttEM 1.1. For any h > 0 and any g > O, there exist matrices, A and C such that every element of A C I is numerically less than X, whereas there are elements of CA I which equal u in magnitude. In other words, C could be a good right-hand inverse of A but a poor lefthand inverse. Let

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