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Sylvester's law of inertia states that the number of positive, negative and zero eigenvalues of Hermitian matrices is preserved under congruence transformations. The same is true of generalized Hermitian definite eigenvalue problems, in which the two matrices are allowed to undergo different congruence transformations, but not for the indefinite case. In this paper we investigate the possible change in inertia under congruence for generalized Hermitian indefinite eigenproblems, and derive sharp bounds that show the inertia of the two individual matrices often still provides useful information about the eigenvalues of the pencil, especially when one of the matrices is almost definite. A prominent application of the original Sylvester's law is in finding the number of eigenvalues in an interval. Our results can be used for estimating the number of real eigenvalues in an interval for generalized indefinite and nonlinear eigenvalue problems. [ABSTRACT FROM AUTHOR]

Abstract: e give an method that computes, for any tree T and interval , how many eigenvalues of T lie within the interval. Our method is based on Sylvester’s Law of Inertia. We use our algorithm to show that the nonzero eigenvalues of a caterpillar are simple. It follows that caterpillars having b back nodes, where , are not integral. We also show that among the regular caterpillars C(b,k) formed by adjoining k legs to each of b back nodes, all positive roots are in the interval , and C(b,k) is not integral if . [ABSTRACT FROM AUTHOR]

Abstract: This paper deals with some inertia theorems in Euclidean Jordan algebras. First, based on the continuity of eigenvalues, we give an alternate proof of Kaneyuki’s generalization of Sylvester’s law of inertia in simple Euclidean Jordan algebras. As a consequence, we show that the cone spectrum of any quadratic representation with respect to a symmetric cone is finite. Second, we present Ostrowski–Schneider type inertia results in Euclidean Jordan algebras. In particular, we relate the inertias of objects and in a Euclidean Jordan algebra when or , where and denote Lyapunov and Stein transformations, respectively. [Copyright &y& Elsevier]

Abstract: In a recent paper [C.R. Johnson, S. Furtado, A generalization of Sylvester’s law of inertia, Linear Algebra Appl. 338 (2001) 287–290], Sylvester’s law of inertia is generalized to any matrix that is ∗-congruent to a diagonal matrix. Such a matrix is called unitoid. In the present paper, an alternative approach to the subject of unitoidness is offered. Specifically, Sylvester’s law of inertia states that a Hermitian n × n matrix of rank r with inertia (p, q, n − r) is ∗-congruent to the direct sumIt is demonstrated herein that a unitoid matrix A of rank r is ∗-congruent to a direct sum of diagonal blocks of the formtogether with the zero block 0I n−r . Moreover, the ϕ’s together with the multiplicities p and q are specified in terms of the eigenvalues and eigenvectors of A † A ∗, where A † is the Moore–Penrose inverse of A. [Copyright &y& Elsevier]