Norms on complex matrices induced by random vectors [r-libre/3283]

Morneau-Guérin, Frédéric (2024). Norms on complex matrices induced by random vectors [review of the book from Chávez, Ángel; Garcia, Stephan Ramon, & Hurley, Jackson]. zbMATH Open.

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Item Type:	Book Reviews
Refereed:	Yes
Status:	Published
Abstract:	This paper introduces a family of norms on the space M_n of n x n complex matrices. These norms arise from a probabilistic framework as they are induced by random vectors whose entries are independent and identically distributed (iid) real-valued random variables with sufficiently many moments. Initially, these norms are defined on complex Hermitian matrices as symmetric functions of their (necessarily real) eigenvalues. This contrasts with Schatten p-norms, which are defined in terms of singular values. To be more specific, these random vector norms do not arise from the machinery of symmetric gauge functions. Rather, they are generalizations of the complete homogeneous symmetric (CHS) polynomial norms introduced in Aguilar, K., Chávez, Á., Garcia, S. R., & Volčič, J. (2022). Norms on complex matrices induced by complete homogeneous symmetric polynomials. \textit{Bulletin of the London Mathematical Society, 54}(6), 2078-2100. The paper is organized as follows: In Section 1, the preliminary concepts and notation are covered. Following this, the main result, which is lengthy and highly technical in nature, is stated. Norms arising from familiar distributions (namely Gamma random variables, Normal random variables, Uniform random variables, Laplace random variables, Bernoulli random variables, Finite discrete random variables, Poisson random variables, and Pareto random variables) are examined in Section 2. Various examples and applications are also provided, including a powerful generalization of Hunter’s positivity theorem for the complete homogeneous symmetric polynomials. The proof of the main result, which involves a wide range of topics, such as cumulants, Bell polynomials, partitions, and Schur convexity, is contained in Section 3. The paper concludes with a list of open questions.
Additional Information:	© FIZ Karlsruhe GmbH 2024. ALL RIGHTS RESERVED.
Official URL:	https://zbmath.org/7741833
Depositor:	Morneau-Guérin, Frédéric
Owner / Manager:	Frédéric Morneau-Guérin
Deposited:	03 Jun 2024 13:08
Last Modified:	06 Jan 2025 15:06

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