On the Submultiplicativity of Matrix Norms Induced by Random Vectors

Abstract

In a recent article, Chávez, Garcia and Hurley introduced a new family of norms $\Vert \cdot {\Vert }_{\mathbf{X},d}$ on the space of $n\times n$ complex matrices which are induced by random vectors $\mathbf{X}$ having finite $d$-moments. Therein, the authors asked under which conditions the norms induced by a scalar multiple of $\mathbf{X}$ are submultiplicative. In this paper, this question is completely answered by proving that this is always the case, as long as the entries of $\mathbf{X}$ have finite $p$-moments for $p=\max \{2+\epsilon ,d\}$.

14 pages

Cited by in corpus (1)

• Hunter's positivity theorem and random vector norms