Condition numbers and their condition numbers

Various normwise relative condition numbers that measure the sensitivity of matrix inversion and the solution of linear systems are characterized. New results are derived for the cases where two common, noninduced matrix norms are used, and where different vector norms are used for the domain and range of the matrix. Condition numbers that respect the structure of symmetric problems are also analyzed. The sensitivity of the condition number itself is then investigated, and we obtain sharp examples of Demmel's general result that for certain problems in numerical analysis 'the condition number of the condition number is the condition number.' Finally, upper bounds are derived for the sensitivity of componentwise condition numbers.