abs_vol20_pp95-102. 28KB Jun 04 2011 12:05:49 AM

ELA

REACHABILITY MATRICES AND CYCLIC MATRICES

∗

AUGUSTO FERRANTE† _{AND HARALD K. WIMMER}‡

Abstract. We study reachability matrices R(A, b) = [b, Ab, . . . , An−1_{b], whereAis ann}×_n

matrix over a fieldKandbis inKn

. We characterize those matrices that are reachability matrices for some pair (A, b). In the case of a cyclic matrixAand ann-vector of indeterminatesx, we derive a factorization of the polynomial det R(A, x)

.

Key words. Reachability matrix, Krylow matrix, cyclic matrix, nonderogatory matrix, com-panion matrix, Vandermonde matrix, Hautus test.

AMS subject classifications.15A03, 15A15, 93B05.

∗_{Received by the editors September 23, 2009. Accepted for publication February 11, 2010.}

Han-dling Editor: Daniel Szyld.

†_{Dipartimento di Ingegneria dell’Informazione, Universit`a di Padova, I-35131 Padova, Italy}

(augusto@dei.unipd.it).

‡_{Mathematisches Institut, Universit¨at W¨urzburg, D-97074 W¨urzburg, Germany}

(wimmer@mathematik.uni-wuerzburg.de).

Electronic Journal of Linear Algebra ISSN 1081-3810 A publication of the International Linear Algebra Society Volume 20, pp. 95-102, February 2010