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arXiv · 2402.08173

On the Submultiplicativity of Matrix Norms Induced by Random Vectors

Ludovick Bouthat
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Abstract

In a recent article, Ch\'avez, Garcia and Hurley introduced a new family of norms $\|\cdot\|_{\mathbf{X},d}$ 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 $p=\max\{2+\varepsilon,d\}$.

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Ludovick Bouthat. 2024-02-13. On the Submultiplicativity of Matrix Norms Induced by Random Vectors. https://doi.org/10.1007/s11785-024-01518-0

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