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Zbl 1108.15015
Glasby, S.P.
The {\sc Meat}-{\sc axe} and $f$-cyclic matrices. (English)
[J] J. Algebra 300, No. 1, 77-90 (2006). ISSN 0021-8693

Let $M(d,F)$ denote the algebra of $d\times d$ matrices over a field $F$, and denote by $m_X(t)$ and $c_X(t)$ the minimal and the characteristic polynomials of $X\in M(d,F)$. We call $X$ an $f$-cyclic matrix if $f$ is an irreducible factor of $m_X(t)$ which does not divide $c_X(t)/m_X(t)$. The paper presents a version of the {\sc Meat}-{\sc axe} algorithm that uses $f$-cyclic matrices. One advantage of $f$-cyclic matrices is that they unify and generalize previous work of {\it R. A. Parker} [Computational group theory, Proc. Symp., Durham/Engl. 1982, 267--274 (1984; Zbl 0555.20001); Lond. Math. Soc. Lect. Note Ser. 249, 215--228 (1998; Zbl 0915.20006)], {\it D. F. Holt} and {\it S. Rees} [J. Aust. Math. Soc., Ser. A 57, No.~1, 1--16 (1994; Zbl 0833.20021)], {\it G. Ivanyos} and {\it K. Lux} [Exp. Math. 9, No.~3, 373--381 (2000; Zbl 0966.16028)], {\it P. M. Neumann} and {\it C. E. Praeger} [J. Lond. Math. Soc., II. Ser. 52, No.~2, 263--284 (1995; Zbl 0839.15011); Ohio State Univ. Math. Res. Inst. Publ. 8, 291--300 (2001; Zbl 0999.20005)]. The greater abundance of $f$-cyclic matrices may lead to an improved probability/complexity analysis of the {\sc Meat}-{\sc axe}. The difficulties that occur when the Schur index exceeds one are explored.
[Rodica Covaci (Cluj-Napoca)]

MSC 2000:: *15A30 Algebraic systems of matrices

Keywords: irreducible; reducible; algorithm; {\sc Meat}-{\sc axe}

Citations: Zbl 0555.20001; Zbl 0915.20006; Zbl 0833.20021; Zbl 0966.16028; Zbl 0839.15011; Zbl 0999.20005

Cited in: Zbl pre05610504

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