Comparison of two norms of matrices Original Research Article

Any complex n × n matrix A satisfies the inequality double vertical bar A double vertical bar 1 ≤ n 1/2 double vertical bar A double vertical bard where short parallel.short parallel1 is the trace norm and short parallel.short paralleld is the norm defined by image, where B is the set of orthonormal bases in the space of n × 1 matrices. The present work is devoted to the study of matrices A satisfying the identity: double vertical barAdouble vertical bar1 = n1/2 double vertical bar A double vertical bar d This paper is a first step towards a characterization of matrices satisfying this identity. Actually, a workable characterization of matrices subject to this condition is obtained only for n = 2. For n = 3, a partial result on nilpotent matrices is presented. Like our previous study (J. Dazord, Linear Algebra Appl. 254 (1997) 67), this study is a continuation of the work of M. Marcus and M. Sandy (M. Marcus and M. Sandy, Linear and Multilinear Algebra 29 (1991) 283). Also this study is related to the work of R. Gabriel on classification of matrices with respect to unitary similarity (see R. Gabriel, J. Riene Angew, Math. 307/308 (1979) 31; R. Gabriel, Math. Z. 200 (1989) 591).