AIMS lecture notes on numerical analysis by Olver P.J.

By Olver P.J.

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When defined, matrix 3/15/06 36 c 2006 Peter J. Olver addition is commutative, A + B = B + A, and associative, A + (B + C) = (A + B) + C, just like ordinary addition. A scalar is a fancy name for an ordinary number — the term merely distinguishes it from a vector or a matrix. For the time being, we will restrict our attention to real scalars and matrices with real entries, but eventually complex scalars and complex matrices must be dealt with. We will consistently identify a scalar c ∈ R with the 1 × 1 matrix ( c ) in which it is the sole entry, and so will omit the redundant parentheses in the latter case.

Nevertheless, the ability to confidently handle all types of linear systems is a basic prerequisite for further progress in the subject. In contemporary applications, particularly those arising in numerical solutions of differential equations, in signal and image processing, and elsewhere, the governing linear systems can be huge, sometimes involving millions of equations in millions of unknowns, challenging even the most powerful supercomputer. So, a systematic and careful development of solution techniques is essential.

A permutation matrix is a matrix obtained from the identity matrix by any combination of row interchanges. In particular, applying a row interchange to a permutation matrix produces another permutation matrix. The following result is easily established. 9. A matrix P is a permutation matrix if and only if each row of P contains all 0 entries except for a single 1, and, in addition, each column of P also contains all 0 entries except for a single 1. 3/15/06 56 c 2006 Peter J. Olver In general, if a permutation matrix P has a 1 in position (i, j), then the effect of multiplication by P is to move the j th row of A into the ith row of the product P A.

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