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ON DESCARTES’ RULE OF SIGNS FOR MATRIX POLYNOMIALST HOMAS R. C AMERON , PANAYIOTIS J. P SARRAKOS
Submitted to Operators and Matrices
Abstract. We present a generalized Descartes’ rule of signs for self-adjoint matrix polynomials
whose coefficients are either positive or negative definite, or null. In particular, we conjecturethat the number of real positive (negative) eigenvalues of a matrix polynomial is bounded aboveby the product of the size of the matrix coefficients and the number of definite sign alternations(permanences) between consecutive coefficients. Our main result shows that this generalization
holds under the additional assumption that the matrix polynomial is hyperbolic. In addition,
we prove individual cases where the matrix polynomial is diagonalizable by congruence, or of
degree three or less. The full proof of our conjecture is an open problem; we discuss analytic
and algebraic approaches for solving this problem and ultimately, what makes this open problem
non-trivial. Finally, we prove generalizations of two famous extensions of Descartes’ rule: If all
eigenvalues are real then the bounds in Descartes’ rule are sharp, and the number of real positive
and negative eigenvalues have the same parity as the associated bounds in Descartes’ rule.

1. Introduction
The theory of matrix polynomials has been strongly influenced by its applications
to differential equations and vibrating systems. In fact, vibrating systems motivated thefirst works devoted primarily to matrix polynomials: one by Frazer, Duncan, and Collar
in 1938 [10] and the other by Lancaster in 1966 [14]. In addition, the theory of matrix
polynomials has been influenced by results from matrix theory and complex analysis.
The canonical set of Jordan chains defined in [12] are a natural generalization of a cycleof generalized eigenvectors. A generalized Rouché’s theorem is presented in [11] andis then used to prove a generalized Pellet’s theorem for matrix polynomials in [17]. The
generalized Pellet’s theorem has been used to give sharp bounds on the spectrum of
unitary matrix polynomials [6] and in the development of approximation methods forthe eigenvalues of a matrix polynomial [4, 5, 18].
These generalizations have greatly influenced the study of matrix polynomials andtheir spectra. It is in this spirit that we present a generalization of Descartes’ rule ofsigns for matrix polynomials. Descartes rule was first published in 1637 [8], but the firstwidely recognized rigorous proof of the rule was not given until 1742 by de Gua [3].
Since then, many proofs of Descartes’ rule and its extensions have been presented [2,3, 7, 20]. Today, we understand Descartes’ rule as follows:
Mathematics subject classification (2010): 15B57, 47B65, 47H60, 47J10.
Keywords and phrases: Descartes’ rule of signs, matrix polynomials, eigenvalues, Hermitian matrices,
definite matrices.

1• A real scalar polynomial has no more positive roots than alternations of signs
between two consecutive coefficients.
• A real scalar polynomial has no more negative roots than permanences of signs
between two consecutive coefficients.
By a matrix polynomial, we mean a polynomial in a complex variable with matrix
coefficients. More specifically, a matrix polynomial of degree m and size n is given byP(λ ) = Am λ m + Am−1 λ m−1 + · · · + A1 λ + A0 ,Am 6= 0,(1)

where the coefficients Ai ∈ Cn×n and λ is a complex variable. We assume that thematrix polynomial is regular, that is, det P(λ ) is not identically zero. A finite eigenvalueof the matrix polynomial P(λ ) is any scalar µ ∈ C such that det P(µ) = 0 . For eacheigenvalue µ ∈ C, a corresponding eigenvector v ∈ Cn is any nonzero vector suchthat P(µ)v = 0 . We note that the matrix polynomial P(λ ) has an infinite eigenvalue
if and only if the leading coefficient Am is singular. Any further discussion of infinite
eigenvalues is beyond the scope of this article and hereafter all eigenvalues are assumed
to be finite.
Throughout this article, we require that the coefficients of P(λ ) are self-adjoint(i.e., Hermitian) and positive or negative definite, or null. We adapt the convention
from [3] for counting alternations and permanences of signs of P(λ ). If consecutive
coefficients Ai+1 and Ai are both positive or negative definite, then the pair (Ai+1 , Ai )contributes 1 permanence and 0 alternation of signs. If one is positive definite andthe other is negative definite, then the pair (Ai+1 , Ai ) contributes 1 alternation and 0
permanence of signs. For null coefficients, we count as follows:• For alternations, every coefficient Ai = 0 is considered to be the same sign as
Ai+1 . Note that this is the same as ignoring the null coefficients.
• For permanences, the sign of the coefficient Ai = 0 is considered to be opposite
to the sign of Ai+1 . This is the same as counting the alternations of P(−λ ).
Henceforth, we denote by z+ (P) the number of positive eigenvalues of P(λ ) andby z− (P) the number of negative eigenvalues. We let S denote the set of all self-adjoint
matrix polynomials whose coefficients are either positive or negative definite, or null.
Furthermore, for each P(λ ) ∈ S, we use α(P) to denote the number of alternations ofsigns of P(λ ) and π(P) to denote the number of its permanences of signs.
The outline of this article is as follows: In Section 2, we show that a generalized
Descartes’ rule holds for hyperbolic and diagonalizable matrix polynomials in S. We
note that in this section we take an analytic approach using derivatives to build inductive
arguments to obtain our results. Whereas, in Section 3, we take an algebraic approachto show that this generalization holds for all matrix polynomials in S of degree three or
less. We conjecture that this generalization holds for all matrix polynomials in S anddiscuss what makes this open problem non-trivial. Finally, in Section 4, we generalize
two famous extensions of Descartes’ rule for matrix polynomials.

22. An Analytic Approach
The numerical range of a matrix polynomial is the set
W (P) = {µ ∈ C : x∗ P(µ)x = 0 for some x ∈ S },(2)

where S denotes the unit sphere in Cn , that is,
S = {x ∈ Cn : x∗ x = 1}.
It is important to note that W (P) contains all the eigenvalues of P(λ ). In addition,
if P(λ ) = λ I − A , then W (P) reduces to the field of values of the matrix A [13].
Finally, we call the matrix polynomial P(λ ) hyperbolic if W (P) is a bounded subset
of R . Note that if P(λ ) is hyperbolic then its coefficients must be self-adjoint and theleading coefficient is either positive definite or negative definite [15, Theorem 2.3]. We
let H ⊂ S denote the set of all hyperbolic matrix polynomials whose coefficients areeither positive or negative definite, or null.
Suppose that P(λ ) is hyperbolic and denote by λ1 (x) ≤ λ2 (x) ≤ · · · ≤ λm (x) the
roots of x∗ P(λ )x , where x ∈ S . The setΛ j = {λ j (x) : x ∈ S}
is called the j th spectral zone of P(λ ). Note that each spectral zone, Λ j , is a closed
bounded interval (possibly degenerate) [α j , β j ] on the real line. In addition, it follows
from [16, Theorem 31.5] that β j ≤ α j+1 , for j = 1, 2, . . . , m−1 , and from [15, Theorem
3.1] that each spectral zone contains exactly n eigenvalues of P(λ ).
0Now, let λ10 (x) ≤ λ20 (x) ≤ . . . ≤ λm−1(x) denote the roots of x∗ P0 (λ )x , wherex ∈ S . Then, by Rolle’s theorem, we have λ j (x) ≤ λ j0 (x) ≤ λ j+1 (x), which implies thatthe spectral zones of P0 (λ ) satisfy
Λ0j = {λ j0 (x) : x ∈ S} ⊆ [α j , β j+1 ],L EMMA 1.

j = 1, 2, . . . , m − 1.

Let P(λ ) ∈ H . Then,z+ (P0 ) ≥ z+ (P) − n.

(3)

Proof. Let s+ (P) denote the number of spectral zones of P(λ ) that lie in R+ . By
the discussion preceding the theorem statement, it follows thats+ (P0 ) ≥ s+ (P) − 1.

(4)

If a spectral zone contains the origin, then it must be the degenerate interval [0] as
the coefficient matrices cannot be indefinite. In particular, if A0 6= 0 , then we have0∈
/ W (P); otherwise, if A0 = 0 and Ak is the trailing coefficient of P(λ ), then F(λ ) =λ −k P(λ ) satisfies 0 ∈/ W (F) and W (P) = W (F) ∪ {0} . Therefore, since each spectral
zone contains exactly n eigenvalues and no spectral zone contains both positive andnegative eigenvalues, the result follows by multiplying both sides of (4) by n .
3Now, we follow the proof of de Gua for Descartes’ rule to prove our main result.
More precisely, we use the next lemma to construct a matrix polynomial from P(λ )that has exactly one less alternation than P(λ ), which allows us to establish a proof by
induction on the number of alternations of signs.
L EMMA 2. Let P(λ ) ∈ S. Suppose that the consecutive coefficients Ak and Ak−1
are of opposite sign and define F(λ ) = λ −k P(λ ). Then,
G(λ ) = λ k+1 F 0 (λ )has exactly one less alternation than P(λ ).
Proof. Note thatG(λ ) = −kP(λ ) + λ P0 (λ )= (m − k)Am λ m + · · · + (k − k)Ak λ k + · · · + (0 − k)A0 .
It is clear from the above equation that the counting of alternations from P(λ ) to G(λ )can differ only in the (k + 1), k , and (k − 1) indexed coefficients. Furthermore, whencounting alternations in G(λ ), the null coefficient (k − k)Ak is considered of the same
sign as [(k + 1) − k]Ak+1 = Ak+1 . The result follows by considering the cases where
Ak+1 and Ak are of opposite sign and the same sign, and noting that in either case G(λ )has precisely one less alternation than P(λ ).
Several remarks are in order: First, Lemma 2 holds for all matrix polynomials in
S, whereas Lemma 1 only holds for matrix polynomials in H ⊂ S. Second, given the
hypothesis of Lemma 2, Ak cannot be a trailing coefficient of P(λ ) since Ak and Ak−1
are of opposite sign. Therefore, the numerical range of F(λ ) satisfies
W (F) = W (P) \ {0}.
Here we are extending the definition in (2) in a natural way to hold for the rational
matrix function F(λ ). It follows that if P(λ ) is hyperbolic, then so too is G(λ ).
Finally,z+ (F) = z+ (P) and z+ (G) = z+ (F 0 ),and it follows from Lemma 1 that if P(λ ) is hyperbolic, thenz+ (G) ≥ z+ (P) − n.

(5)

We are now ready to prove our main result.
T HEOREM 3.

Let P(λ ) ∈ H . Then,z+ (P) ≤ n · α(P) and z− (P) ≤ n · π(P).

(6)

Proof. We prove the first assertion and note that the second assertion follows bychanging P(λ ) into P(−λ ).

4The following is a proof by induction on the number of alternations of signs. If
α(P) = 0 , then x∗ P(λ )x is a real scalar polynomial with zero alternations of signs,
for all x ∈ S . Therefore, by Descartes’ rule, it follows that W (P) ∩ R+ = 0/ . Thus,
z+ (P) = 0 and (6) holds.
Suppose, then, that the result holds for all matrix polynomials in H with k ≥ 0alternations. Let P(λ ) ∈ H such that α(P) = k + 1 . Applying Lemma 2, we constructG(λ ) ∈ H such that α(G) = k . Hence, by the induction hypothesis, z+ (G) ≤ n·α(G).
Furthermore, by (5), we havez+ (P) ≤ z+ (G) + n≤ n · α(G) + n= n(k + 1) = n · α(P).
Therefore, by the principle of mathematical induction, (6) holds for all P(λ ) ∈ H .
Informally, we note that the proofs of Descartes’ rule for scalar polynomials havefallen into two categories: analytic and algebraic. Typically, analytic proofs rely on thegeometry of curves and extrema to obtain information, or derivatives to build inductive
arguments. In contrast, algebraic proofs rely on the factorization of the polynomial
and the properties of an ordered field [3]. For this reason, we say that our proof of
Theorem 3 is analytic in nature.
We are interested in extending the set of matrix polynomials for which our main
result (6) holds. To that end, we note that (6) holds for all P(λ ) ∈ S such that α(P) = 0 .
Furthermore, given any P(λ ) ∈ S with α(P) = k + 1 we can use Lemma 2 to form aG(λ ) ∈ S such that α(G) = k . It follows from the proof of Theorem 3 that if we
can extend the set of matrix polynomials for which (3) holds, then we can extend theset of matrix polynomials for which (6) holds. For instance, (3) clearly holds for all
scalar polynomials in S, due to Rolle’s theorem. Therefore, (6) holds for all real scalar
polynomials and it follows that Descartes’ rule is a special case of a much broader
theory. Similarly, the following proposition implies that (6) holds for all P(λ ) ∈ S thatare diagonalizable by congruence.
P ROPOSITION 4.

If P(λ ) ∈ S is diagonalizable by congruence then (3) holds.

Proof. If P(λ ) is diagonalizable by congruence, then there exists a nonsingular
M ∈ Cn×n such that MP(λ )M ∗ = D(λ ), where D(λ ) is a diagonal matrix polynomialwith real coefficients. Note that D0 (λ ) = MP0 (λ )M ∗ .
It follows that the eigenvalues of P(λ ) are the roots of the diagonal entries ofD(λ ), denoted by dii (λ ), and the eigenvalues of P0 (λ ) are the roots of the diagonal
entries of D0 (λ ), denoted by dii0 (λ ), for i = 1, 2, . . . , n. Moreover, as a consequence of
Rolle’s theorem, between any two distinct real positive roots of dii (λ ) there is a realpositive root of dii0 (λ ).
Finally, for i = 1, 2, . . . , n , let ri and ri0 denote the number of real positive roots ofdii (λ ) and dii0 (λ ), respectively. It follows that ri0 ≥ ri − 1 , for i = 1, 2, . . . , n. Hence,z+ (D0 ) ≥ z+ (D) − n and, therefore, z+ (P0 ) ≥ z+ (P) − n.

5

We conjecture that (6), in fact, holds for all matrix polynomials in S. However,as the following example illustrates, (3) does not hold for all matrix polynomials in
S. For this reason, we must consider other methods for extending the set of matrix
polynomials for which (6) holds.
E XAMPLE 5. Consider the matrix polynomial:
 5
λ − 6λ 4 + 15λ 3 − 20λ 2 + 14λ − 4
λ 4 − 5λ 3 + 9λ 2 − 7λ + 2
P(λ ) =.
λ 4 − 5λ 3 + 9λ 2 − 7λ + 2
λ 5 − 7λ 4 + 19λ 3 − 25λ 2 + 16λ − 4Note that P(λ ) ∈ S and α(P) = 5 . Hence, (6) holds trivially (see Lemma 6). However,P(λ ) has 8 real positive eigenvalues and P0 (λ ) has only 4 real positive eigenvalues.
Therefore, (3) does not hold.
3. An Algebraic Approach
The concluding remarks made in Section 2 serve as the impetus of this section,
where we prove that (6) holds for all matrix polynomials in S of degree three or less.
We begin with a lemma that establishes the result for a specific number of alternations.
L EMMA 6.

Let P(λ ) ∈ S. Then, z+ (P) ≤ n·α(P) provided that α(P) ∈ {0, 1, m} .

Proof. Note that the case α(P) = m is trivial since the total number of eigenvaluesof P(λ ) is equal to mn, and the case α(P) = 0 was already covered in the proof of
Theorem 3 and did not rely on the matrix polynomial being hyperbolic.
Therefore, we only need to consider the case where α(P) = 1 . Then, for eachx ∈ S , the real scalar polynomial x∗ P(λ )x has exactly one alternation and, by Descartes’
rule, precisely one positive root. We may assume that the leading coefficient of P(λ )is positive definite. Otherwise, the matrix polynomial −P(λ ) has a positive definite
leading coefficient and the same number of alternations and positive eigenvalues asP(λ ). Therefore, there exists a, b ∈ R+ such that a < b , P(a) is negative definite,P(b) is positive definite, and for each x ∈ S , x∗ P(λ )x has exactly one root in (a, b). It
follows from [16, Theorem 30.6] that P(λ ) admits the factorization
P(λ ) = P+ (λ )(λ I − Z),
where P+ (λ ) is invertible for all λ ∈ (a, b) and all eigenvalues of Z are contained in(a, b). Thus, P(λ ) has exactly n positive eigenvalues, and the result follows.
The fact that (6) holds for linear and quadratic matrix polynomials in S follows
readily from the previous lemma. With a little more work, we also prove the cubic case.
T HEOREM 7.
m ∈ {1, 2, 3} .

Let P(λ ) ∈ S. Then, (6) holds provided that P(λ ) has degree

Proof. For degree m = 1 , or m = 2 , the only cases to consider are covered by
Lemma 6.

6

If m = 3, then in addition to appealing to Lemma 6, we must also show that theresult holds for α(P) = 2 and π(P) = 2 , where the latter result follows from the firstby changing P(λ ) into P(−λ ).
Suppose that α(P) = 2 and for the sake of contradiction assume that the purely
imaginary number it , where t ∈ R \ {0} , lies in W (P). Then, there exists a vectorx ∈ S such that−i(x∗ A3 x)t 3 − (x∗ A2 x)t 2 + i(x∗ A1 x)t + (x∗ A0 x) = 0.
Note that if A2 = 0 , then in order to satisfy the above equation it follows that A0 = 0and, therefore, there is at most 1 alternation, which contradicts α(P) = 2 . Hence, we
may assume that A2 6= 0 and it follows thatx∗ A0 x x∗ A1 x=
= t 2 > 0.
x∗ A2 x x∗ A3 x
However, this implies that there are 0 or 3 alternations, both of which contradict
α(P) = 2 . Therefore, W (P) does not intersect the imaginary axis, except, possibly,
at the origin. That being said, if A0 = 0 , then the matrix polynomial Q(λ ) obtainedby dividing P(λ ) by λ is quadratic with 2 alternations, and the result follows from
Lemma 6. Therefore, we may assume that P(λ ) does not have a zero coefficient after
its trailing coefficient, and W (P) does not intersect the imaginary axis anywhere.
Thus, W (P) consists of two parts: ΩL in the open left half-plane of C and ΩR in
the open right half-plane. As a real cubic polynomial with 2 alternations x∗ P(λ )x has
at least one root in ΩL , but ΩR may be empty. If ΩR is empty, then z+ (P) = 0 , andthe result follows.
Suppose, then, that ΩR is non-empty. By the continuity of the roots of x∗ P(λ )xwith respect to x ∈ S , it follows that the number of roots of the polynomials x∗ P(λ )x inΩR is constant and equal to 2 . Therefore, we may construct a simple closed rectifiable
curve Γ in the open left half-plane such that for each x ∈ S , x∗ P(λ )x has no roots on
the boundary of Γ and exactly 1 root inside Γ . By [16, Theorem 26.19], it follows thatP(λ ) can be factored in the formP(λ ) = PL (λ )PR (λ ),where PL (λ ) has n eigenvalues in ΩL and PR (λ ) has 2n eigenvalues in ΩR . Therefore,z+ (P) ≤ n · α(P) when α(P) = 2 , and the result follows.
The proof of Theorem 7 is algebraic in nature since it relies on the factorizationof the matrix polynomial. In this fashion, we could attempt to extend the set of matrix
polynomials for which (6) holds one degree at a time. For all possible degrees, this will
require an inductive strategy and some analogue of Segner’s lemma [3]:If a scalar polynomial is multiplied by (λ − γ), with γ > 0, then its number ofalternations of signs increases by at least 1 . If it is multiplied by (λ + γ), then its
number of permanences of signs increases by at least 1 .
Since the coefficients of the matrix polynomials in S do not form a field we havelittle hope in applying this strategy.
7

In summary, both analytic and algebraic strategies for proving Descartes’ rule haveserious complications in the generalization for matrix polynomials. The strategy forproving Theorem 3 relies on Lemma 1, which does not hold for all matrix polynomialsin S (see Example 5). Furthermore, there is no inductive strategy for working with thefactorization results employed in this section. For this reason, we consider the extensionof the set of matrix polynomials for which (6) holds to be a non-trivial open problem.
Furthermore, in order to significantly extend this set (recall that our conjecture is (6)holds for all matrix polynomials in S), we will likely need to find a completely different
approach.
4. Extensions of Descartes’ Rule
There are two famous extensions of Descartes’ rule that, throughout history, havebecome synonymous with the original rule. The first extension is that if all roots ofa real scalar polynomial are real, then the bounds of Descartes’ rule are sharp. This
statement is an easy to prove corollary of Descartes’ rule. In addition, the converseof this statement holds for polynomials that have no null coefficients, except, possibly,
after their trailing coefficient.
The second extension states that the parities of the number of positive roots andalternations of signs are equal, and the parities of the number of negative roots and thepermanence of signs are equal. We refer to this extension as Fourier’s rule since it wasnoted and proven in the 1820 dissertation of Joseph Fourier [3]. It is worth noting that
Fourier’s rule holds independently of Descartes’ rule.
In this section, we prove generalizations of both extensions for matrix polynomialsin S. When proving both results, we assume, without loss of generality, that P(λ )has no null coefficients after its trailing coefficient. Otherwise, the matrix polynomial
F(λ ) obtained by dividing P(λ ) by a suitable power λ k has no null coefficients after
its trailing coefficient and satisfies
z+ (P) = z+ (F), z− (P) = z− (F), α(P) = α(F), and π(P) = π(F).
T HEOREM 8.

Let P(λ ) ∈ S. If (6) holds and all eigenvalues of P(λ ) are real,thenz+ (P) = n · α(P) and z− (P) = n · π(P),i.e., the bounds in (6) are sharp.
Proof. Note that, for i = m − 1, m − 2, . . . , 0 , each pair of consecutive coefficients(Ai+1 , Ai ) can contribute at most one alternation or permanence. Therefore, we haveα(P) + π(P) ≤ m. For the sake of contradiction, suppose that all eigenvalues of P(λ )are real and z+ (P) < n · α(P) or z− (P) < n · π(P). Since P(λ ) has no null coefficients
after its trailing coefficient, it follows that
nm = z+ (P) + z− (P) < n(α(P) + π(P)),
which implies that α(P) + π(P) > m and contradicts what we know to be true.

8The converse of Theorem 8 holds for all matrix polynomials in S that have nonull coefficients, except, possibly, after their trailing coefficient. Indeed, let P(λ ) ∈ S
and let k denote the index of the last null coefficient after the trailing coefficient, wherek = −1 if P(λ ) has no null coefficient. Then, α(P)+π(P) = m−(k +1) and assuming
the bounds in (6) are sharp we have z+ (P) + z− (P) = n (m − (k + 1)). Since P(λ ) hasn(k + 1) zero eigenvalues, it follows that all nm eigenvalues of P(λ ) are real.
T HEOREM 9. Let P(λ ) ∈ S. Then n · α(P) and z+ (P) are of the same parity, as
are n · π(P) and z− (P).
Proof. We prove the first assertion and note that the second assertion follows bychanging P(λ ) into P(−λ ). Recall that we assume, without loss of generality, thatP(λ ) has no null coefficients after its trailing coefficient.
Let p(λ ) = det P(λ ). Then p(λ ) is a real scalar polynomial, which factors asp(λ ) = cp1 (λ )p2 (λ ), where p1 (λ ) is monic polynomial with no positive roots, andp2 (λ ) is a monic polynomial with only positive roots. Clearly the degree of p2 (λ ) isequal to z+ (P) and c = det Am . Since the complex roots of a real scalar polynomial
come in conjugate pairs, it follows that p1 (0) > 0 . In addition, note that p(0) = det A0
and p2 (0) > 0 if and only if z+ (P) is even. It follows that the signs of det A0 and
det Am are equal if and only if z+ (P) is even.
Since a sequence of signs contains an even number of alternations if and only if its
extremities are equal, it follows that α(P) is even if and only if both matrices A0 and
Am are of the same sign. If α(P) is odd, then the signs of det A0 and det Am are equal
if and only if n is even. Thus, z+ (P) is even if and only if n · α(P) is even, and theresult follows.
5. Conclusion
The set S of self-adjoint matrix polynomials whose coefficients are positive or
negative definite, or null, provides a framework for the generalization of Descartes’
rule of signs and its extensions. In Section 2, we prove our generalized Descartes’
rule, which is stated formally in (6), holds for all hyperbolic and diagonalizable matrix
polynomials in S. Then, in Section 3, we prove that (6) holds for all matrix polynomialsin S of degree three or less. Also, two generalized extensions of Descartes’ rule arepresented in Theorem 8 and Theorem 9.
We conjecture that (6) holds for all matrix polynomials in S and, in Sections 2
and 3, we discuss the difficulties surrounding a complete solution to this problem. In
particular, we conclude that a significant extension of the set of matrix polynomials forwhich (6) holds will require a completely different approach.
A natural first step is to attempt to generalize Lemma 1 to quasi-hyperbolic matrix
polynomials [1]. However, even this extension appears to be non-trivial. For instance,even if P(λ ) is strictly isospectral to a diagonal matrix polynomial D(λ ), we are notaware of a relationship between P0 (λ ) and D0 (λ ), nor are we aware of a relationshipbetween the definiteness of P(λ ) and the definiteness of D(λ ). Moreover, much of
the current results for quasi-hyperbolic matrix polynomials deal with the positive or
negative type of the eigenvalues and not their sign.
9That being said, this is one of many possible directions to consider for extending
the set of matrix polynomials in S for which (6) holds. We note that the set S consistsof matrix polynomials that arise naturally in the study of differential equations andvibrating systems [9, 10, 12, 14, 19]. Hence, future investigation of our generalized
Descartes’ rule and its extensions, such as those noted in [7], may lead to additional
interesting results and useful applications.
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Thomas R. Cameron, Davidson College
e-mail: thcameron@davidson.edu
Panayiotis J. Psarrakos, National Technical University of Athens
e-mail: ppsarr@math.ntua.gr
Corresponding Author: Thomas R. Cameron

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