Linear Independence and Rank. The Simple Image Set

Abstract

Efficient methods for checking linear independence and for finding the coefficients of linear dependence were presented in Chap. 3. That chapter also contains results on anomalies of linear independence. For these and other reasons various alternative concepts of linear independence have been studied. In most cases they would be equivalent to the above mentioned definition if formulated in linear algebra; however, in max-algebra they are nonequivalent. Two other concepts of independence are studied in this chapter: strong linear independence and Gondran-Minoux independence. Particular attention is paid to these concepts and their mutual comparison in the setting of square matrices, that is, to a comparison of regularity, strong regularity and Gondran-Minoux regularity. All three properties can be checked in polynomial time. In the case of the latter two this is due to their strong relation to the assignment problem. However, it is not known whether strong linear independence and Gondran-Minoux independence can be checked in polynomial time. As a by-product of strong regularity it is proved that the nonempty set of simple images of a max-linear mapping for a generic class of matrices coincides with the interior of the eigenspace.