Properties of the weighted logarithmic matrix norms

In this paper, we are concerned with the properties of the weighted logarithmic matrix norms. A relation between the elliptic logarithmic matrix norm and the weighted logarithmic matrix norm is given. Based on Lyapunov equations, two weighted logarithmic matrix norms are constructed which are less than 1-logarithmic matrix norm and ∞-logarithmic matrix norm, respectively. Then, an iterative scheme is presented to obtain the logarithmically ϵ-efficient matrix norm. Numerical examples are given to illustrate the results.