Sign-non-singular matrices and matrices with unbalanced determinant in symmetrised semirings Original Research Article

The operations circled plus and circle times operator are defined by acircled plusb=max{a,b},acircle times operatorb=a+b over the set of reals extended by −∞. The columns a(1),…,a(n) of an n×n matrix A are said to be linearly dependent in max-algebra if image ∑jset membership, variantUcircled plusλjcircle times operatora(j)=∑jset membership, variantVcircled plusλjcircle times operatora(j)holds for some λ1,…,λnset membership, variantR;U,V≠empty set︀;U∩V=empty set︀;Uunion or logical sumV={1,…,n}. We prove that there is a close relationship between sign-nonsingular (SNS) (0,1,−1) matrices and matrices with unbalanced determinant in symmetrised semirings. Given a matrix A we then show how to construct a (0,1,−1) matrix à such that A has columns linearly dependent in max-algebra if and only if à is not SNS. Also, it follows that if the system Acircle times operatorx=Bcircle times operatorx has a nontrivial solution then image is not SNS, where C=AΘB (here Θ stands for the subtraction in the symmetrised semiring). As another corollary we have a new, independent proof that the problems of checking whether a matrix is SNS and that of deciding whether a digraph contains a cycle of even length, are polynomially equivalent.