Sign patterns of the Schwarz matrices and generalized Hurwitz polynomials

Abstract

The direct and inverse spectral problems are solved for a wide subclass of the class of Schwarz matrices. A connection between the Schwarz matrices and the so-called generalized Hurwitz polynomials is found. The known results due to H. Wall and O. Holtz are briefly reviewed and obtained as particular cases.

Self-interlacing polynomials

“…Remark 2.3. Another, more complicated, proof of Theorem 2.2 can be found in the technical report [19] (see Theorem 4.3 there).…”

“…In [9] (see also [18] and references there) there were introduced tridiagonal matrices of the form: whose characteristic polynomials belong to SI I if b 1 > 0 (it was was established implicitly without mentioning of self-interlacing polynomials). In [9], the author also proved that for any polynomial p ∈ SI I , there exists a unique matrix of the form (1.3) with b 1 > 0 whose characteristic polynomial is p. Clearly, for b 1 < 0, we deal with the polynomials in the class SI II .…”

“…All other references can be found there. Some results of this work were mentioned in the technical report [19], but here we substantially simplified most of the proofs and provided additional properties of self-interlacing polynomials.…”

“…Implicitly, the self-interlacing polynomials seem to appear first time in [4, Lemma 2.6] (in its first edition) where a necessary condition for real polynomials to be self-interlacing was established (see Theorem 3.4 of the present work). In [9] (see also [18] and references there) there were introduced tridiagonal matrices of the form:…”