upper bound on matrix exponential

Question

I need an upper bound for the following term norm(I-e^(Ax)) in which A is an n*n real matrix ,x is a scalar and I is the unit matrix. is there any upper bound that is zero at x=0? if not what is the smallest upper bound that you find?

Answer 1

Using the power series expansion for the matrix exponential,

$||I -e^{Ax}|| = || Ax + \frac{(Ax)^2}{2!} + \frac{(Ax)^3}{3!} + ... || \leq e^{||A|||x|} -1$