Generalized Hurwitz Matrices, Generalized Euclidean Algorithm, and Forbidden Sectors of the Complex Plane
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Generalized Hurwitz Matrices, Generalized Euclidean Algorithm, and Forbidden Sectors of the Complex Plane Olga Holtz1,2 · Sergey Khrushchev3 · Olga Kushel4
Received: 23 June 2015 / Accepted: 24 September 2015 © Springer-Verlag Berlin Heidelberg 2016
Abstract Given a polynomial f (x) = a0 x n + a1 x n−1 + · · · + an with positive coefficients ak , and a positive integer M ≤ n, we define an infinite generalized Hurwitz matrix HM ( f ) := (a M j−i )i, j . We prove that the polynomial f (z) does not vanish in the sector
z ∈ C : | arg(z)|
0) is stable if and only if all leading principal minors of its Hurwitz matrix H2 ( f ) up to order n are positive. Decades after Routh–Hurwitz, Asner [2] and Kemperman [16] independently realized that the Routh–Hurwitz criterion can be restated in terms of the total nonnegativity of the Hurwitz matrix. Moreover, the Hurwitz matrix of a stable polynomial admits a simple factorization into totally non-negativ
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