How many vectors generate maximal cyclic subspaces?
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.
scientific article; zbMATH DE number 1903542