Expected Conditioning

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Abstract
A perturbation analysis based on probabilistic arguments is developed for a range of problems in numerical linear algebra, including well-determined and over-determined linear systems. Condition matrices and condition numbers are determined for the expected value of the actual condition number of a problem. These enable attainable lower and upper bounds on the expected condition properties of a matrix to be given, independent of any particular linear system. These estimates are much more reliable than those derived from conventional norm condition numbers, and are shown to reveal features which the latter cannot. The expected condition analysis has desirable properties under scaling transformations which is not the case for the norm condition analysis. It is shown that an optimal (or natural) scaling can be associated with any matrix which moreover is readily computed. This enables the equilibration of a matrix to be carried out. Because this process is a uniquely defined projection it always enables the best conditioned of all the possible equilibrated matrices to be determined.

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