Sign patterns of the Schwarz matrices and generalized Hurwitz polynomials

Electronic Journal of Linear Algebra 1081-3810 Sign patterns of the Schwarz matrices and generalized Hur witz polynomials Mikhail Tyaglov Follow this and additional works at: https://repository.uwyo.edu/ela Recommended Citation - http://math.technion.ac.il/iic/ela MIKHAIL TYAGLOV† AMS subject classifications. 15A29, 47A75, 47B36, 26C10. 1. Introduction. In this work, we consider the matrices of the form (1.1) bk ∈ R\{0}  −b0  −b1  0  ...   0  0 which are usually called the Schwarz matrices. Note that in [ 16 ] Schwarz also considered matrices whose (1, 1)-entry is zero while (n, n)-entry is not. Sometimes such matrices are called the Schwarz matrices as well, see [ 8 ]. In this paper, we solve direct and inverse problems for such matrices with certain sign patterns. At first, let us note that such matrices (more exactly, their characteristic polynomials for the case of bk > 0, k = 0, . . . , n − 1) were considered first time by Wall in his study [ 18 ] on Hurwitz stable polynomials, i.e. the polynomials with zeroes in the open left half-plane. Schwarz [ 16 ] extended Wall’s result to real nonzero bk, and applied ∗Received by the editors on January 3, 2012. Accepted for publication on November 3, 2012 Handling Editor: Carlos Fonseca. †Shanghai Jiao Tong University, Department of Mathematics, 800 Dong Chuan Road, 200240, Shanghai, China (). The work was performed at Technische Universit¨at Berlin and at Shanghai Jiao Tong University and was supported by the European Research Council under the European Union’s Seventh Framework Programme (FP7/2007-2013)/ERC grant agreement no. 259173. 216 M. Tyaglov purely matrix methods to study matrices similar to the matrices of the form (1.1). Thus these matrices should be called rather Wall-Schwarz matrices. The matrices (1.1) are well-studied from the matrix theory point of view (see e.g. [ 3, 15, 4, 5, 6, 7 ] and references there). Here we use the method due to Wall [ 18, 19 ] to solve the inverse spectral problem, and use our results on the generalized Hurwitz polynomials to solve the direct spectral problem for the Schwarz matrices (1.1) with certain sign patterns. The case of all bk positive was considered by Wall [ 18 ] and later by Schwarz [ 16 ] and many other authors. The case of all bk negative was studied by Holtz [ 13 ]. Moreover, the case b1 > 0, b2, . . . , bn−1 < 0 was considered in [ 1 ]. The present work deals with certain intermediate cases between positive and negative bk, that is, we study a number of sign patterns for bk. Our results include, of course, the results by Wall [ 18 ] and by Holtz [ 13 ]. Such a generalization becomes possible due to relationships between the entries bk of the matrix (1.1) and the Hurwitz determinants (see (2.5)) of its characteristic polynomial.Those relationships were found by Wall in [ 18 ]. The Hurwitz stable polynomials studied by Wall and the polynomials appeared in the paper by Holtz [ 13 ] turned out to be connected by a one-to-one correspondence [ 17 ] (see Theorem 4.7 of the present paper). In [ 17 ] it was also discovered that there exists a general class of polynomials whose distribution of zeroes can be described by signs of their Hurwitz determinants, which includes Hurwitz stable polynomials as a subclass, the so-called generalized Hurwitz polynomials. So the Wall’s relationship mentioned above gives a possibility to study all matrices of the form (1.1) whose characteristic polynomials are generalized Hurwitz, and to solve the direct and inverse spectral problems for them. Recall also that Schwarz in his paper [ 16 ] combined the same relationship and the Routh-Hurwitz theorem to study the location of the spectra of the matrices (1.1) with respect to the imaginary axis. Thus the present work generalizes results by Wall [ 18 ], Holtz [ 13 ], and Bebiano and Providˆencia [ 1 ], and specifies some results by Schwarz [ 16 ]. The paper is organized as follows. In Section 2 we discuss some results by Wall [ 18 ] and obtain a consequence that may be new (see Theorem 2.2). In this section we also introduce some constructions and auxiliary results that will be of use in what follows. For the sake of the reader’s convenience, Section 3 is devoted to known results on the direct and inverse problems for the matrices of the form (1.1). We present the results by Wall, Schwarz, and Holtz in the form that allows the reader to easily compare their results with results of the next sections and to understand what new was done by the author. In Section 4 we recall some basic facts on the generalized Hurwitz polynomials established in [ 17 ]. In Section 5 we prove our main theorems on the direct and inverse problems for Schwarz matrices with certain sign patterns. Note that we state our results in the same form as Schwarz did in [16, Satz 5] (see Sign patterns of the Schwarz matrices and generalized Hurwitz polynomials 217 Theorem 3.4 of the present work). Fi (...truncated)