A new weighted metric: the relative metric I

Abstract The M-relative distance, denoted by ρM 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 ρM 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.  2002 Elsevier Science (USA). All rights reserved.