Linear Algebra and its Applications 374 (2003) 247–253 www.elsevier.com/locate/laa Equivalence constants for certain matrix norms Bao Qi Feng Department of Mathematical Sciences, Kent State University, Tuscarawas Campus, New Philadelphia, OH 44663-9403, USA Received 25 January 2002; accepted 9 May 2003 Submitted by M. Goldberg Abstract We obtain optimal equivalence constants for certain families of matrix norms. Our results generalize inequalities of Goldberg [Linear and Multilinear Algebra 21 (1987) 173]. © 2003 Elsevier Inc. All rights reserved. Keywords: p-Norms; (p,q)-Norms; p-Operator norms; (p,q)-Operator norms; Equivalence constants In what follows all scalars are complex. Let l n p be the Banach space of all n-tuples x = (x 1 ,...,x n ) ∈ C n equipped with the norm |x | p =         n ∑ j =1 |x j | p  1/p (1  p< ∞), max 1j n |x j | (p =∞). The vector space C m×n of all complex m × n matrices A = (a ij ) can be given many natural norms. We focus on the (p,q)-norm, |A| pq =  n  j =1  m  i =1 |a ij | p  q/p  1/q (1  p, q  ∞), and the (p,q)-operator norm, ‖A‖ pq = max{|Ax | q :|x | p  1} (1  p, q  ∞). In the special case that p = q, we call these norms the p-norm and p-operator norm, respectively. E-mail address: bfeng@tusc.kent.edu (B.Q. Feng). 0024-3795/$ - see front matter  2003 Elsevier Inc. All rights reserved. doi:10.1016/S0024-3795(03)00616-5