Condition numbers for functions of matrices (Q1802649)

Usual, mixed, and structured condition numbers \(F\) for functions of matrices are considered. A representation of the Fréchet derivatives is obtained for functions \(F(\lambda)\) in the form \[ {F(u)-F(v)\over u-v}=\sum^ m_{k=1}\int^ 1_ 0G_ k(s,u)H_ k(s,v)ds. \] Such formulas are derived for some elementary functions: power, exponential, logarithm, and sine and cosine, which lead to some old and sometimes new expressions for condition numbers of the corresponding matrix maps. These expressions help to obtain estimates on condition numbers. They can also be used for calculation of condition numbers.