TY - JOUR

T1 - Testing the irreducibility of nonsquare Perron-Frobenius systems

AU - Avin, C.

AU - Borokhovich, M.

AU - Haddad, Y.

AU - Kantor, E.

AU - Lotker, Z.

AU - Parter, M.

AU - Peleg, D.

PY - 2014/12

Y1 - 2014/12

N2 - The Perron-Frobenius (PF) theorem provides a simple characterization of the eigenvectors and eigenvalues of irreducible nonnegative square matrices. A generalization of the PF theorem to nonsquare matrices, which can be interpreted as representing systems with additional degrees of freedom, was recently presented in [1]. This generalized theorem requires a notion of irreducibility for nonsquare systems. A suitable definition, based on the property that every maximal square (legal) subsystem is irreducible, is provided in [1], and is shown to be necessary and sufficient for the generalized theorem to hold. This note shows that irreducibility of a nonsquare system can be tested in polynomial time. The analysis uses a graphic representation of the nonsquare system, termed the constraint graph, representing the flow of influence between the constraints of the system.

AB - The Perron-Frobenius (PF) theorem provides a simple characterization of the eigenvectors and eigenvalues of irreducible nonnegative square matrices. A generalization of the PF theorem to nonsquare matrices, which can be interpreted as representing systems with additional degrees of freedom, was recently presented in [1]. This generalized theorem requires a notion of irreducibility for nonsquare systems. A suitable definition, based on the property that every maximal square (legal) subsystem is irreducible, is provided in [1], and is shown to be necessary and sufficient for the generalized theorem to hold. This note shows that irreducibility of a nonsquare system can be tested in polynomial time. The analysis uses a graphic representation of the nonsquare system, termed the constraint graph, representing the flow of influence between the constraints of the system.

KW - Algorithms

KW - Irreducibility

KW - Perron-Frobenius theorem

UR - http://www.scopus.com/inward/record.url?scp=84904274056&partnerID=8YFLogxK

U2 - 10.1016/j.ipl.2014.05.004

DO - 10.1016/j.ipl.2014.05.004

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AN - SCOPUS:84904274056

SN - 0020-0190

VL - 114

SP - 728

EP - 733

JO - Information Processing Letters

JF - Information Processing Letters

IS - 12

ER -