How many vectors generate maximal cyclic subspaces?
type: Item
instance of: scholarly article
title: How many vectors generate maximal cyclic subspaces?
zbMATH Open document ID: 1028.15014
DOI: 10.1016/S1071-5797(02)00015-1
author: William C. Waterhouse
published in: Finite Fields and their Applications
publication date: 2003-05-04T00:00:00Z
review text: The problem of generating the cyclic subspaces of the \(d\times d\)-matrices with minimal polynomials of degree \(m\leq d\) is analyzed. It is taken into account that in the case of infinite spaces for such one matrix \(A\) there exist \(d\)-tuples \(\nu\) which generate the cyclic subspace spanned on \((\nu, A\nu, A^2\nu,\dots)\) of dimension \(m\) [cf. \textit{A. S. Householder}, The theory of matrices in numerical analysis (1964; Zbl 0161.12101); \textit{T. Y. Li}, \textit{Z. Zhang} and \textit{T. Wang}, Linear Algebra Appl. 252, 221-259 (1997; Zbl 0870.65030)]. In the paper another situation, occurring of finite fields [cf. \textit{P. M. Neumann} and \textit{C. E. Praeger}, J. Lond. Math. Soc., II. Ser. 52, No. 2, 263-284 (1995; Zbl 0839.15011)] is analyzed. It is shown that for a \(d\times d\)-matrix \(B\) over finite field with \(q\) elements the probability that \(d\)-tuple generates a maximal cyclic subspace for \(B\) is at least \(e^{-2}\log(q)[\log(q)+ \log(d)]\). Some examples of such order of magnitude are considered.
Mathematics Subject Classification ID: 15B33
Mathematics Subject Classification ID: 15A03
Mathematics Subject Classification ID: 15A24
zbMATH DE Number: 1903542
zbMATH Keywords: minimal polynomials
zbMATH Keywords: maximal cyclic subspaces
zbMATH Keywords: finite spaces
zbMATH Keywords: infinite spaces
zbMATH Keywords: finite fields
reviewed by: A. A. Bogush
MaRDI profile type: Publication
cites work: Q5543004
cites work: Determining the structure of the Jordan normal form of a matrix by symbolic computation
cites work: Q3765876
cites work: Cyclic Matrices Over Finite Fields
cites work: Polynomials with Irreducible Factors of Specified Degree
full work available at URL: s1071-5797(02)00015-1
OpenAlex ID: W2051552231
description: scientific article; zbMATH DE number 1903542