A new weighted metric: the relative metric I

The M-relative distance, denoted by ρ_M is a generalization of the p-relative distance introduced in [R.-C. Li, SIAM J. Matrix Anal. Appl. 19 (1998) 956-982]. We establish necessary and sufficient conditions under which ρ_M is a metric. In two special cases we derive complete characterizations of this metric. We also present a way of extending the results to metrics sensitive to the domain in which they are defined and find some connections to previously studied metrics. An auxiliary result of independent interest is an inequality related to Pittenger's inequality in Section 4.