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SOME COUNTING QUESTIONS FOR MATRIX PRODUCTS

Part of: Polynomials and matrices Basic linear algebra

Published online by Cambridge University Press:  09 October 2023

MUHAMMAD AFIFURRAHMAN Open the ORCID record for MUHAMMAD AFIFURRAHMAN [Opens in a new window]
MUHAMMAD AFIFURRAHMAN*
Affiliation:
School of Mathematics and Statistics, University of New South Wales, Sydney NSW 2052, Australia
*
e-mail: m.afifurrahman@unsw.edu.au
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Abstract

Given a set X of $n\times n$ matrices and a positive integer m, we consider the problem of estimating the cardinalities of the product sets $A_1 \cdots A_m$, where $A_i\in X$. When $X={\mathcal M}_n(\mathbb {Z};H)$, the set of $n\times n$ matrices with integer elements of size at most H, we give several bounds on the cardinalities of the product sets and solution sets of related equations such as $A_1 \cdots A_m=C$ and $A_1 \cdots A_m=B_1 \cdots B_m$. We also consider the case where X is the subset of matrices in ${\mathcal M}_n(\mathbb {F})$, where $\mathbb {F}$ is a field with bounded rank $k\leq n$. In this case, we completely classify the related product set.

Keywords

matrixmatrix equationinteger matrixmatrix product

MSC classification

Primary: 11C20: Matrices, determinants
Secondary: 15A24: Matrix equations and identities

Information

Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 110 , Issue 1 , August 2024 , pp. 32 - 43
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Copyright
© The Author(s), 2023. Published by Cambridge University Press on behalf of Australian Mathematical Publishing Association Inc.

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Footnotes

The author is supported by an UNSW Tuition Fee Scholarship and Australian Research Council Grant DP200100355.

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