Approximation of Matrix-Valued Functions

It is shown that if $f( z ) = \sum _{i = 0}^\infty a_i z^i $ (the series having radius of convergence R) then for any $n \times n$ matrix A with spectral radius less than R and any norm $\| \cdot \|$ on the space of $n \times n$ matrices \[ \left\| f ( A ) - \sum_{i = 0}^k a_i A_i \right\| \leq \frac{1}{( k + 1 )!} \max\limits_{s \in [ 0,1 ]} \left\| A^{k + 1} f^{( k + 1 )} ( sA ) \right\|. \] 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 $n \times n$ matrix-valued functions.