APPROXIMATION OF MATRIX-VALUED FUNCTIONS

It is shown that if f(z) = SIGMA(i=0)infinity a(i)z(i) (the series having radius of convergence R) then for any n x n matrix A with spectral radius less than R and any norm \\.\\ on the space of n x n matrices [GRAPHICS] It is also shown that expressions for the error in numerical integration rules can be generalized to matrix-valued functions. The main point of the paper is that in these two cases it is not necessary to increase the bound by a factor depending on n when generalizing an inequality for scalar-valued functions to n x n matrix-valued functions.