Taylor's Theorem for Matrix Functions with Applications to Condition Number Estimation

We derive an explicit formula for the remainder term of a Taylor polynomial of a matrix function. This formula generalizes a known result for the remainder of the Taylor series for an analytic function of a complex scalar. We investigate some consequences of this result, which culminate in new upper bounds for the level-1 and level-2 condition numbers of a matrix function in terms of the pseudospectrum of the matrix. Numerical experiments show that, although the bounds can be pessimistic, they can be computed almost three orders of magnitude faster than the standard methods for the $1$-norm condition number of $f(A) = A^t$. This makes the upper bounds ideal for a quick estimation of the condition number whilst a more accurate (and expensive) method can be used if further accuracy is required