Spectrally Perron Polynomials and the Cauchy-Ostrovsky Theorem

In this note, we simplify the statements of theorems attributed to Cauchy and Ostrovsky and give proofs of each theorem via combinatorial and nonnegative matrix theory. We also show that each simple sufficient condition in each statement is also necessary in its respective case. In addition, we introduce the notion of a spectrally Perron polynomial and pose a problem that appeals to a wide mathematical audience.


Introduction
Because the roots of a polynomial coincide with the eigenvalues of its companion matrix, many classical results on the geometry of polynomials can be obtained with relative ease via matricial methods: for 15 instance, Wilf [10] used the Perron-Frobenius theroem to derive Cauchy's bound on the maximum modulus of any given root and Bell [1] used the Gershgorin Theorem to derive the same bound along with other bounds and a result due to Walsh (see [5, §5.6] for a bounds obtained via matrix norms).
The purpose of this work is to give proofs of results attributed to Cauchy and Ostrovsky via combinatorial matrix theory and nonnegative matrix theory. In the process of doing so, we simplify the 20 statement of each of these results by giving a single, simple sufficient condition that is shown to be necessary (respectively) in each theorem. In addition, we introduce the notion of a spectrally Perron polynomial and pose a problem that appeals to a wide mathematical audience.

Notation & Background
A real matrix is called nonnegative (positive) if it is entrywise nonnegative (respectively, positive) matrix. 25 A directed graph (or simply digraph) Γ = (V, E) consists of a finite, nonempty set V of vertices, together with a set E ⊆ V × V of arcs. For an n-by-n matrix A, the digraph of A, denoted by Γ = Γ (A), has vertex set V = {1, . . . , n} and arc set A digraph Γ is called strongly connected if for any two distinct vertices i and j of Γ, there is a walk in Γ from i to j (following [2], we consider every vertex of V as strongly connected to itself). A 30 strong digraph is primitive if the greatest common divisor of all its cycle-lengths is one, otherwise it is imprimitive.  [4,6,8]. 10 Given an n-by-n matrix A, the characteristic polynomial of A, denoted by χ A , is defined by χ A = det (tI − A). The companion matrix C = C p of a monic polynomial p(t) = t n + n k=1 c k t n−k is the n-by-n matrix defined by where c = [c n−1 · · · c 1 ]. It is well-known that χ C = p. Notice that C is irreducible if and only if c n = 0. Finally, we call a polynomial p (weakly) spectrally Perron if its companion matrix is (weakly) spec-15 trally Perron.

The Cauchy-Ostrovosky Theorem
Before we state the proofs, we introduce some brief notation. In this section, p(t) = t n −c 1 t n−1 −· · ·−c n , where c k ≥ 0, for every k. show that the sufficient condition is also necessary.

Theorem 3.1 (Cauchy). The polynomial p is weakly spectrally Perron if and only if d = 0.
Proof. Denote by J n (λ) the n-by-n Jordan block with eigenvalue λ. If d = 0, then at least one of the elements of I is positive, i.e., at least one of the elements of C is positive. If is the largest index such that c > 0, then The result now follows by applying the Perron-Frobenius theorem for irreducible nonnegative matrices [5,Theorem 8.4.4] to the matrixĈ. For the converse, we proceed by the contrapositive: to that end, if d = 0, then C = J n (0) and p is clearly not weakly spectrally Perron.
The following result, attributed to Ostrovsky, provides a sufficient condition that ensures that p is 5 spectrally Perron; we give a proof via nonnegative matrix theory and show that the sufficient condition is also necessary. Proof. If d = 1, then at least one of the elements of C is positive. We distinguish the following cases: i) c n = 0. In this case C is irreducible and since d = 1, note that at least two elements of C are positive 10 (otherwise d = n). Moreover, C is primitive since the greatest common divisor of all cycle-lengths of Γ (C) is one. Hence, p is spectrally Perron. ii) c n = 0. In this case, C is of the form (3.1) and the result follows by applying the exact argument in the previous case to the irreducible matrixĈ.
To prove necessity, we proceed via the contrapositive; to that end, and without losing generality, 15 assume that d > 1 (the case d = 0 was handled in the proof of Cauchy's Theorem). We distinguish the following cases: i) c n = 0. In this case, C is irreducible and ρ = ρ (C) ∈ σ (C), but C has d eigenvalues of modulus ρ (see, e.g., [5,Corollary 8.4.6(c)]). Thus, p is weakly spectrally Perron but not spectrally Perron. ii) c n = 0. In this case, C is of the form (3.1) and the result follows by applying the exact argument in the 20 previous case to the irreducible matrixĈ.
From the above, p is spectrally Perron if and only if d = 1.

Implications for Further Research
A real matrix A is called eventually nonnegative (postive) if there is a positive integer k such that A k is nonnegative (respectively, positive). If p is a monic polynomial and C p is eventually nonnegative, then 25 p is weakly spectrally Perron [3]; if C p is eventually positive, then p is spectrally Perron [4,6,8]. However, as the following example indicates, the companion matrix of a spectrally Perron polynomial need not be eventually nonnegative.  The previous example leads to the following problem.