INERTIA SETS OF SYMMETRIC SIGN PATTERN MATRICES

1　IntroductionIn qualitative and combinatorial matrix theory,we study properties ofa matrix basedon combinatorial information,such as the signs of entries in the matrix.A matrix whoseentries are from the set{ + ,-,0 } is called a sign pattern matrix ( or sign pattern,or pat-tern) .We denote the setof all n× n sign pattern matrices by Qn.For a real matrix B,sgn( B) is the sign pattern matrix obtained by replacing each positive( respectively,negative,zero) entry of B by+ ( respectively,-,0 )…

INERTIA SETS OF SYMMETRIC SIGN PATTERN MATRICES

A sign pattern matrix is a matrixwhose entries are from the set {+ ,- ,0}. The symmetric sign pattern matrices that require unique inertia have recently been characterized. The purpose of this paper is to more generally investigate the inertia sets of symmetric sign pattern matrices. In particular, nonnegative fri-diagonal sign patterns and the square sign pattern with all + entries are examined. An algorithm is given for generating nonnegative real symmetric Toeplitz matrices with zero diagonal of orders n≥3 which have exactly two negative eigenvalues. The inertia set of the square pattern with all + off-diagonal entries and zero diagonal entries is then analyzed. The types of inertias which can be in the inertia set of any sign pattern are also obtained in the paper. Specifically, certain compatibility and consecutiveness properties are established.