Second Logarithmic Derivative of a Complex Matrix in the Chebyshev Norm

Abstract

The second logarithmic derivative $\mu^{(2)}_{\infty}[A]$ of a complex n x n matrix A in the Chebyshev norm is defined as the second right derivative of $\| \Phi(t) \|_{\infty} \,=\, \| e^{A t} \|_{\infty}$ at $t=0$, where $\| \cdot \|_{\infty}$ denotes the operator norm corresponding to the norm $\| \cdot \|_{\infty}$ in $\CC^n$. The obtained formula is illustrated by a numerical example. The result may be of interest in applications such as stability theory.