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Norms on complex matrices induced by random vectors

Part of: Rings with polynomial identity General theory of linear operators Basic linear algebra

Published online by Cambridge University Press:  23 December 2022

Ángel Chávez ,
Stephan Ramon Garcia  and
Jackson Hurley
Ángel Chávez*
Affiliation:
Department of Mathematics and Statistics, Pomona College, 610 North College Avenue, Claremont, CA 91711, USA e-mail: stephan.garcia@pomona.edu jacksonwhurley@gmail.com
Stephan Ramon Garcia
Affiliation:
Department of Mathematics and Statistics, Pomona College, 610 North College Avenue, Claremont, CA 91711, USA e-mail: stephan.garcia@pomona.edu jacksonwhurley@gmail.com
Jackson Hurley
Affiliation:
Department of Mathematics and Statistics, Pomona College, 610 North College Avenue, Claremont, CA 91711, USA e-mail: stephan.garcia@pomona.edu jacksonwhurley@gmail.com
*
e-mail: angel.chavez@pomona.edu
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Abstract

We introduce a family of norms on the $n \times n$ complex matrices. These norms arise from a probabilistic framework, and their construction and validation involve probability theory, partition combinatorics, and trace polynomials in noncommuting variables. As a consequence, we obtain a generalization of Hunter’s positivity theorem for the complete homogeneous symmetric polynomials.

Keywords

Normsymmetric polynomialpartitiontracepositivityconvexityexpectationcomplexificationtrace polynomialprobability distribution

MSC classification

Primary: 47A30: Norms (inequalities, more than one norm, etc.) 15A60: Norms of matrices, numerical range, applications of functional analysis to matrix theory 16R30: Trace rings and invariant theory
Type
Article
Information
Canadian Mathematical Bulletin , Volume 66 , Issue 3 , September 2023 , pp. 808 - 826
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Copyright
© The Author(s), 2022. Published by Cambridge University Press on behalf of The Canadian Mathematical Society

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Footnotes

S.R.G. was partially supported by the NSF (Grant No. DMS-2054002).

References

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