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SOME COUNTING QUESTIONS FOR MATRIX PRODUCTS
Published online by Cambridge University Press: 09 October 2023
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.
MSC classification
Primary:
11C20: Matrices, determinants
Secondary:
15A24: Matrix equations and identities
- Type
- Research Article
- Information
- Copyright
- © The Author(s), 2023. Published by Cambridge University Press on behalf of Australian Mathematical Publishing Association Inc.
Footnotes
The author is supported by an UNSW Tuition Fee Scholarship and Australian Research Council Grant DP200100355.
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