Hurwitz-Type Matrix Polynomials

• Hurwitz-type matrix polynomials are defined by an even–odd decomposition that produces a rational matrix function with a finite Stieltjes continued fraction featuring positive definite coefficients.
• They are analyzed through moment data and block Hankel matrices derived from Hermitian Markov parameters, with Dyukarev–Stieltjes parameters ensuring positive definiteness.
• Under additional commutativity conditions, explicit Bezoutian factorizations and inertia arguments confirm Hurwitz stability, linking classical stability criteria to the matrix framework.

Searching arXiv for relevant papers on Hurwitz-type matrix polynomials and closely related stability criteria. Hurwitz-type matrix polynomials are matrix polynomials

$$f_n(z):=A_0 z^n + A_1 z^{n-1} + \cdots + A_n,\qquad A_k\in\mathbb{C}^{q\times q},\ \det A_0\neq 0,$$

whose even–odd decomposition

$f_n(z)=h_n(z^2)+z\,g_n(z^2)$

induces a rational matrix function admitting a finite continued fraction with positive definite matrix coefficients. In the formulation developed in "On the Hurwitz Stability of Hurwitz-Type Matrix Polynomials" (Choque-Rivero, 15 Jul 2025), the even-degree case requires $g_{2m}(z)h_{2m}^{-1}(z)$ to have that form, while the odd-degree case requires $\frac{1}{z}h_{2m+1}(z)g_{2m+1}^{-1}(z)$ to do so. The subject lies at the intersection of matrix continued fractions, truncated Stieltjes matrix moment problems, orthogonal matrix polynomials, Bezoutians, and Hurwitz stability, and it extends a classical scalar line of thought in which Stieltjes-type fractions, Hankel positivity, and Hurwitz theory are tightly coupled (Choque-Rivero, 15 Jul 2025, Zhan et al., 2019, Dyachenko, 2013).

1. Algebraic form and defining decomposition

Every matrix polynomial of degree $n$ admits the decomposition

$f_n(z)=h_n(z^2)+z\,g_n(z^2),$

with the even and odd coefficient blocks separated according to the parity of $n$. For even degree $n=2m$,

$$h_{2m}(z):=A_0 z^m + A_2 z^{m-1} + \cdots + A_{2m},\qquad g_{2m}(z):=A_1 z^{m-1} + A_3 z^{m-2} + \cdots + A_{2m-1}.$$

For odd degree $n=2m+1$,

$f_n(z)=h_n(z^2)+z\,g_n(z^2)$0

This decomposition is the basic algebraic mechanism behind the theory (Choque-Rivero, 15 Jul 2025).

A matrix polynomial $f_n(z)=h_n(z^2)+z\,g_n(z^2)$1 is called Hurwitz-type if

$f_n(z)=h_n(z^2)+z\,g_n(z^2)$2

admits a finite continued fraction with positive definite matrix coefficients. Likewise, $f_n(z)=h_n(z^2)+z\,g_n(z^2)$3 is Hurwitz-type if

$f_n(z)=h_n(z^2)+z\,g_n(z^2)$4

admits such a representation. The continued fractions are matrix Stieltjes continued fractions associated to extremal solutions of a nondegenerate truncated Stieltjes matrix moment problem on $f_n(z)=h_n(z^2)+z\,g_n(z^2)$5, and the corresponding rational functions are matrix-valued Stieltjes transforms corresponding to positive measures on $f_n(z)=h_n(z^2)+z\,g_n(z^2)$6 (Choque-Rivero, 15 Jul 2025).

In the even case one has

$f_n(z)=h_n(z^2)+z\,g_n(z^2)$7

and in the odd case

$f_n(z)=h_n(z^2)+z\,g_n(z^2)$8

where the $f_n(z)=h_n(z^2)+z\,g_n(z^2)$9 and $g_{2m}(z)h_{2m}^{-1}(z)$0 are positive definite $g_{2m}(z)h_{2m}^{-1}(z)$1 matrices (Choque-Rivero, 15 Jul 2025).

2. Moment-theoretic data, Hankel matrices, and Dyukarev–Stieltjes parameters

The continued-fraction definition is accompanied by a moment-theoretic description. For an even-degree Hurwitz-type matrix polynomial, the ratio $g_{2m}(z)h_{2m}^{-1}(z)$2 has a Laurent expansion at infinity of the form

$g_{2m}(z)h_{2m}^{-1}(z)$3

and for odd degree the ratio $g_{2m}(z)h_{2m}^{-1}(z)$4 has the analogous expansion. The coefficients $g_{2m}(z)h_{2m}^{-1}(z)$5 are Hermitian Markov parameters (Choque-Rivero, 15 Jul 2025).

From these parameters one forms the block Hankel matrices

$g_{2m}(z)h_{2m}^{-1}(z)$6

Nondegenerate truncated Stieltjes matrix moment feasibility requires $g_{2m}(z)h_{2m}^{-1}(z)$7, and for Hurwitz-type matrix polynomials one has $g_{2m}(z)h_{2m}^{-1}(z)$8 (Choque-Rivero, 15 Jul 2025).

The Dyukarev–Stieltjes parameters are then recovered from Schur complements and inverse Hankel blocks. In particular,

$g_{2m}(z)h_{2m}^{-1}(z)$9

and for $\frac{1}{z}h_{2m+1}(z)g_{2m+1}^{-1}(z)$0,

$\frac{1}{z}h_{2m+1}(z)g_{2m+1}^{-1}(z)$1

These matrices are Hermitian positive definite for a nondegenerate truncated Stieltjes matrix moment problem (Choque-Rivero, 15 Jul 2025).

This moment-theoretic layer places Hurwitz-type matrix polynomials in the same conceptual lineage as the 2019 generalization of classical Hurwitz criteria to matrix polynomials, where Hurwitz stability is tested through positive definiteness of block-Hankel matrices built from matricial Markov parameters and through matricial Stieltjes continued fractions (Zhan et al., 2019). It also aligns with the Herglotz–Nevanlinna viewpoint, where rational functions constructed from even–odd parts are characterized via Laurent coefficients and block-Hankel negativity or positivity conditions (Zhan, 2020).

3. Orthogonal matrix polynomials, second-kind polynomials, and coprimeness

A central structural ingredient is the use of orthogonal matrix polynomials on $\frac{1}{z}h_{2m+1}(z)g_{2m+1}^{-1}(z)$2. Let $\frac{1}{z}h_{2m+1}(z)g_{2m+1}^{-1}(z)$3 be a nonnegative $\frac{1}{z}h_{2m+1}(z)g_{2m+1}^{-1}(z)$4 measure on $\frac{1}{z}h_{2m+1}(z)g_{2m+1}^{-1}(z)$5. A sequence $\frac{1}{z}h_{2m+1}(z)g_{2m+1}^{-1}(z)$6 of $\frac{1}{z}h_{2m+1}(z)g_{2m+1}^{-1}(z)$7 matrix polynomials is left-orthogonal if

$\frac{1}{z}h_{2m+1}(z)g_{2m+1}^{-1}(z)$8

The construction in (Choque-Rivero, 15 Jul 2025) uses two orthogonal families $\frac{1}{z}h_{2m+1}(z)g_{2m+1}^{-1}(z)$9, $n$0 and their second-kind polynomials $n$1, $n$2, which satisfy

$n$3

together with the second-kind integral identities

$n$4

These identities supply the analytic machinery needed to pass from continued fractions and moment data to algebraic properties of the polynomial pair $n$5 (Choque-Rivero, 15 Jul 2025).

The principal result in this direction is the coprimeness theorem: if $n$6 is Hurwitz-type, then $n$7 and $n$8 are right coprime; if $n$9 is Hurwitz-type, then $f_n(z)=h_n(z^2)+z\,g_n(z^2),$0 and $f_n(z)=h_n(z^2)+z\,g_n(z^2),$1 are right coprime. Equivalently, there exist polynomial matrices $f_n(z)=h_n(z^2)+z\,g_n(z^2),$2, $f_n(z)=h_n(z^2)+z\,g_n(z^2),$3 such that

$f_n(z)=h_n(z^2)+z\,g_n(z^2),$4

The proof rests on identities involving $f_n(z)=h_n(z^2)+z\,g_n(z^2),$5, $f_n(z)=h_n(z^2)+z\,g_n(z^2),$6, and the positive definite matrices $f_n(z)=h_n(z^2)+z\,g_n(z^2),$7, together with the explicit representation

$f_n(z)=h_n(z^2)+z\,g_n(z^2),$8

This is one of the points at which the HTM framework departs sharply from scalar Hurwitz theory: coprimeness must be formulated in right- or left-polynomial terms rather than through ordinary scalar gcd arguments (Choque-Rivero, 15 Jul 2025).

The broader matrix-stability literature makes the same issue explicit. In the Herglotz–Nevanlinna formulation of matrix Hurwitz stability, right coprimeness of the even and odd parts is an essential hypothesis in the equivalence between stability and the HN property of the associated rational function (Zhan, 2020).

4. Bezoutians and the commutativity-type condition

The 2025 treatment isolates a coefficient constraint under which the Bezoutian associated with a Hurwitz-type matrix polynomial becomes explicit. For $f_n(z)=h_n(z^2)+z\,g_n(z^2),$9, or $n$0, the condition is

$n$1

with the ranges of $n$2 and $n$3 determined by the parity of $n$4. This is referred to as a commutativity-type condition, and it is used to ensure that certain generalized Bezoutian forms become polynomials in both variables, in the sense of the Anderson–Jury criterion (Choque-Rivero, 15 Jul 2025).

For rational matrix factorizations $n$5, the generalized Bezoutian form is

$n$6

and it is polynomial in $n$7 if and only if $n$8. In the Hurwitz-type setting the associated form is

$n$9

The paper introduces auxiliary forms $n=2m$0 and $n=2m$1 and proves that, under the commutativity-type condition, $n=2m$2, $n=2m$3, and $n=2m$4 are polynomials in $n=2m$5 and $n=2m$6 (Choque-Rivero, 15 Jul 2025).

The ensuing factorization expresses these forms through finite block Hankel matrices $n=2m$7, $n=2m$8, symmetrizers $n=2m$9, Vandermonde-type vectors $$h_{2m}(z):=A_0 z^m + A_2 z^{m-1} + \cdots + A_{2m},\qquad g_{2m}(z):=A_1 z^{m-1} + A_3 z^{m-2} + \cdots + A_{2m-1}.$$0, $$h_{2m}(z):=A_0 z^m + A_2 z^{m-1} + \cdots + A_{2m},\qquad g_{2m}(z):=A_1 z^{m-1} + A_3 z^{m-2} + \cdots + A_{2m-1}.$$1, and alternating-sign diagonal blocks $$h_{2m}(z):=A_0 z^m + A_2 z^{m-1} + \cdots + A_{2m},\qquad g_{2m}(z):=A_1 z^{m-1} + A_3 z^{m-2} + \cdots + A_{2m-1}.$$2. For both even and odd degree, $$h_{2m}(z):=A_0 z^m + A_2 z^{m-1} + \cdots + A_{2m},\qquad g_{2m}(z):=A_1 z^{m-1} + A_3 z^{m-2} + \cdots + A_{2m-1}.$$3 is represented as a quadratic form in those block objects. This yields a matrix-structured Bezoutian factorization that feeds directly into an inertia argument (Choque-Rivero, 15 Jul 2025).

A plausible implication is that the commutativity-type hypothesis is less about the existence of Hurwitz-type structure itself than about the availability of a polynomial Bezoutian calculus strong enough to support a direct stability proof. That interpretation is consistent with the explicit counterexample in which Condition C fails but the polynomial is still Hurwitz, discussed below (Choque-Rivero, 15 Jul 2025).

5. Hurwitz stability and adjacent matrix criteria

In the terminology of (Choque-Rivero, 15 Jul 2025), a matrix polynomial $$h_{2m}(z):=A_0 z^m + A_2 z^{m-1} + \cdots + A_{2m},\qquad g_{2m}(z):=A_1 z^{m-1} + A_3 z^{m-2} + \cdots + A_{2m-1}.$$4 is Hurwitz if $$h_{2m}(z):=A_0 z^m + A_2 z^{m-1} + \cdots + A_{2m},\qquad g_{2m}(z):=A_1 z^{m-1} + A_3 z^{m-2} + \cdots + A_{2m-1}.$$5 has all zeros in the open left half-plane. Under the commutativity-type condition, every Hurwitz-type matrix polynomial is Hurwitz. The proof uses the inertia theorem via Bezoutians due to Lerer–Tismenetsky: if $$h_{2m}(z):=A_0 z^m + A_2 z^{m-1} + \cdots + A_{2m},\qquad g_{2m}(z):=A_1 z^{m-1} + A_3 z^{m-2} + \cdots + A_{2m-1}.$$6 is regular and there exists $$h_{2m}(z):=A_0 z^m + A_2 z^{m-1} + \cdots + A_{2m},\qquad g_{2m}(z):=A_1 z^{m-1} + A_3 z^{m-2} + \cdots + A_{2m-1}.$$7 such that

$$h_{2m}(z):=A_0 z^m + A_2 z^{m-1} + \cdots + A_{2m},\qquad g_{2m}(z):=A_1 z^{m-1} + A_3 z^{m-2} + \cdots + A_{2m-1}.$$8

and

$$h_{2m}(z):=A_0 z^m + A_2 z^{m-1} + \cdots + A_{2m},\qquad g_{2m}(z):=A_1 z^{m-1} + A_3 z^{m-2} + \cdots + A_{2m-1}.$$9

then the spectrum of $n=2m+1$0 lies in the upper half-plane. Setting

$n=2m+1$1

and combining the Bezoutian factorization with the positivity of $n=2m+1$2 and $n=2m+1$3, the paper obtains positivity of the relevant Bezoutian and concludes that the spectrum of $n=2m+1$4 lies in the left half-plane (Choque-Rivero, 15 Jul 2025).

This result is adjacent to, but not identical with, earlier matricial Hurwitz criteria. The 2019 work "On generalization of classical Hurwitz stability criteria for matrix polynomials" proves that, under Hermitian Markov-parameter assumptions, a monic matrix polynomial is Hurwitz-stable if and only if the associated Markov sequence is Stieltjes positive definite; concretely, the decisive tests are positive definiteness of the principal block-Hankel matrices and, equivalently, existence of a matricial Stieltjes continued fraction with positive definite blocks (Zhan et al., 2019). The 2020 Herglotz–Nevanlinna approach gives a different characterization: Hurwitz stability is equivalent to HN properties of rational matrix functions built from the even and odd parts, together with right coprimeness and negativity of the relevant zeros (Zhan, 2020).

The 2025 HTM result is therefore best viewed as a specialized stability theorem for a continued-fraction-defined class. It does not replace the broader Hankel or Herglotz–Nevanlinna criteria, but it connects them to a concrete matricial Stieltjes construction and to an explicit Bezoutian factorization (Choque-Rivero, 15 Jul 2025).

6. Extensions, examples, scalar antecedents, and open questions

The paper proposes an extension by completion. Given a monic polynomial $n=2m+1$5 of degree $n=2m+1$6 that is not Hurwitz-type, one seeks

$n=2m+1$7

such that $n=2m+1$8 is Hurwitz-type. If such a $n=2m+1$9 exists, then $f_n(z)=h_n(z^2)+z\,g_n(z^2)$00 is Hurwitz. The proof uses the orthogonal-matrix-polynomial representation

$f_n(z)=h_n(z^2)+z\,g_n(z^2)$01

and the known location results for zeros of $f_n(z)=h_n(z^2)+z\,g_n(z^2)$02, which place the zeros of $f_n(z)=h_n(z^2)+z\,g_n(z^2)$03 in $f_n(z)=h_n(z^2)+z\,g_n(z^2)$04 (Choque-Rivero, 15 Jul 2025). The construction is algorithmic in the sense that it proceeds through moments $f_n(z)=h_n(z^2)+z\,g_n(z^2)$05, second-kind polynomials, symmetry checks for the rational function $f_n(z)=h_n(z^2)+z\,g_n(z^2)$06, positivity of $f_n(z)=h_n(z^2)+z\,g_n(z^2)$07 and $f_n(z)=h_n(z^2)+z\,g_n(z^2)$08, and the recovery of

$f_n(z)=h_n(z^2)+z\,g_n(z^2)$09

Several examples delineate the scope of the theory. Any $f_n(z)=h_n(z^2)+z\,g_n(z^2)$10 Hurwitz-type polynomial of degree $f_n(z)=h_n(z^2)+z\,g_n(z^2)$11,

$f_n(z)=h_n(z^2)+z\,g_n(z^2)$12

is Hurwitz and satisfies Condition C trivially. A $f_n(z)=h_n(z^2)+z\,g_n(z^2)$13 degree-$f_n(z)=h_n(z^2)+z\,g_n(z^2)$14 example obtained from an absolutely continuous distribution on $f_n(z)=h_n(z^2)+z\,g_n(z^2)$15 yields an HTM polynomial $f_n(z)=h_n(z^2)+z\,g_n(z^2)$16 for which Condition C holds; by the main theorem it is Hurwitz, and $f_n(z)=h_n(z^2)+z\,g_n(z^2)$17 is right coprime. By contrast, the paper also constructs a degree-$f_n(z)=h_n(z^2)+z\,g_n(z^2)$18 HTM polynomial $f_n(z)=h_n(z^2)+z\,g_n(z^2)$19 for which Condition C fails, yet $f_n(z)=h_n(z^2)+z\,g_n(z^2)$20 still has all zeros with negative real parts, so the polynomial is Hurwitz (Choque-Rivero, 15 Jul 2025).

That counterexample leads directly to the main open point emphasized in the paper: Condition C is sufficient to make the Bezoutian forms polynomial and to prove Hurwitzness via Bezoutians, but the paper does not claim that all Hurwitz-type matrix polynomials are Hurwitz without Condition C. This remains an open question. A conjecture is also proposed that the determinants of the HTM coefficients $f_n(z)=h_n(z^2)+z\,g_n(z^2)$21 are positive; the paper states that this is supported by examples but not proved (Choque-Rivero, 15 Jul 2025).

The scalar antecedents of the theory are unusually explicit. The matrix continued-fraction conditions generalize the classical Stieltjes continued fractions of the scalar Markov and Stieltjes moment problems, where positivity of the scalar parameters yields Hurwitz polynomials via the Hermite–Biehler theorem (Choque-Rivero, 15 Jul 2025). In the scalar and meromorphic setting, total nonnegativity of an infinite Hurwitz-type matrix $f_n(z)=h_n(z^2)+z\,g_n(z^2)$22 is equivalent to the ratio $f_n(z)=h_n(z^2)+z\,g_n(z^2)$23 being a meromorphic Stieltjes $f_n(z)=h_n(z^2)+z\,g_n(z^2)$24-function, and equivalently to the existence of a unique regular $f_n(z)=h_n(z^2)+z\,g_n(z^2)$25-fraction with nonnegative coefficients (Dyachenko, 2013). That line of work generalizes the Asner–Kemperman description of totally nonnegative Hurwitz matrices of quasi-stable polynomials and supplies the infinite-matrix background against which the matricial HTM class can be read (Dyachenko, 2013).

Taken together, these results present Hurwitz-type matrix polynomials as a distinguished matrix-valued Stieltjes class: they are defined by positive definite finite matrix continued fractions, encoded by Hermitian Markov parameters and positive block Hankel data, realized through orthogonal matrix polynomials and second-kind polynomials, and—under a commutativity-type condition—converted into Hurwitz matrix polynomials by an explicit Bezoutian argument (Choque-Rivero, 15 Jul 2025).