The Perron eigenspace of nonnegative almost skew-symmetric matrices and Levinger s transformation

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1 Linear Algebra and its Applications 360 (003) The Perron eigenspace of nonnegative almost skew-symmetric matrices and Levinger s transformation Panayiotis J. Psarrakos a, Michael J. Tsatsomeros b, a Department of Mathematics, National Technical University, Zografou Campus, Athens 15780, Greece b Mathematics Department, 107 Neill Hall, Washington State University, Pullman, WA , USA Received 18 March 00; accepted 4 April 00 Submitted by C.-K. Li Abstract Let A be a nonnegative square matrix whose symmetric part has rank one. Tournament matrices are of this type up to a positive shift by 1/I. When the symmetric part of A is irreducible, the Perron value and the left and right Perron vectors of L(A, α) = (1 α)a + αa t are studied and compared as functions of α [0, 1/]. In particular, upper bounds are obtained for both the Perron value and its derivative as functions of the parameter α via the notion of the q-numerical range. 00 Elsevier Science Inc. All rights reserved. AMS classification: 15A18; 15A4; 15A60; 05C0 Keywords: Almost skew-symmetric matrix; Perron value; Perron vector; Levinger s transformation; q-numerical range; Tournament 1. Introduction An almost skew-symmetric matrix is a square matrix whose symmetric part has rank one. The initial interest in almost skew-symmetric matrices can be largely attributed to their association with tournament matrices; indeed, if T is a (0,1)-tournament matrix, i.e., T + T t = J I, wherej is the all ones matrix, then T + 1/I is a nonnegative almost skew-symmetric matrix. The discovery that the eigenvalues of Corresponding author. addresses: tsat@math.wsu.edu (M.J. Tsatsomeros), ppsarr@math.ntua.gr (P.J. Psarrakos) /03/$ - see front matter 00 Elsevier Science Inc. All rights reserved. PII: S (0)

2 44 P.J. Psarrakos, M.J. Tsatsomeros / Linear Algebra and its Applications 360 (003) almost skew-symmetric matrices satisfy interesting inequalities forced upon by their structure [4,5,11] is sustaining the interest in these matrices and their properties. In this paper, we consider an entrywise nonnegative almost skew-symmetric matrix A and study the spectral radius of L(A, α) = (1 α)a + αa t as a function of α [0, 1/]. The affine transformation (1 α)a + αa t and, in particular, its spectral radius when A is nonnegative were initially considered by Levinger [7], who showed that the spectral radius is non-decreasing in [0, 1/]. We thus refer to L(A, α) as Levinger s transformation. This result, re-proved and extended by Fiedler [,3], is regarded as one of the few results on the behavior of the spectral radius as a function of matrix elements and has proven useful in many instances. More recently, use of Levinger s transformation is made in [11] in order to study the spectrum of general almost skew-symmetric matrices. In addition, L(A, α) is used to study the shape of the numerical range of nonnegative matrices [10]. It appears that Levinger s transformation is a piece of many puzzles and so we are motivated to continue studying its role. Here, we do it in the context of nonnegative almost skew-symmetric matrices with irreducible symmetric parts. Specifically, we study the rate of change of the spectral radius of L(A, α) as a function of α,aswell as the behavior of the corresponding left and right (Perron) eigenvectors.. Preliminaries Let x,y R n. We call x a unit vector if its Euclidean norm x = 1. The angle between two nonzero vectors x and y is defined to be ( x,y) = cos 1 x t y x y and is measured in [0, π]. Consider an n n real matrix A (denoted by A M n (R)). The spectrum of A is denoted by σ and its spectral radius by ρ = max{ λ : λ σ }. Recall that A can be written as A = S + K, where S = A + At and K = A At are the (real) symmetric part and the (real) skew-symmetric part of A, respectively. We define Levinger s transformation of A by L(A, α) = (1 α)a + αa t, α [0, 1/] and Levinger s function by φ(a,α) = ρ(l(a, α)) = ρ((1 α)a + αa t ), α [0, 1/]. (.1) Notice that the analysis of Levinger s function in [0, 1/] extends naturally to [1/, 1] as L(A, α) t = L(A, 1 α). One can also see that

3 P.J. Psarrakos, M.J. Tsatsomeros / Linear Algebra and its Applications 360 (003) L(A, 0) = A, L(A, 1/) = S and that for every α (0, 1/), L(A, α) = S + (1 α)k. Suppose now that A is entrywise nonnegative (denoted by A 0). Then S 0. Moreover, suppose S is of rank one. This means that σ (S) consists of the eigenvalue 0 with multiplicity n 1 and a simple eigenvalue δ. By the Perron Frobenius theorem (see e.g. [1]), δ = ρ(s) > 0. It follows that there is a nonzero nonnegative vector w R n such that S = ww t and δ = w t w. We refer to a matrix A having the above features as a nonnegative almost skewsymmetric matrix and assume the reader recalls the above associated notation. The variance of a nonnegative almost skew-symmetric matrix A is var = Kw w = wt (K t K)w w t. w Note that the variance of L(A, α) is given by var(l(a, α)) = (1 α) var. It is also clear that φ(a,0) = ρ and φ(a,1/) = ρ(s) = δ. Some more terminology is in order for a nonnegative matrix X (for details and general background see [1]). We refer to ρ(x) σ(x) as the Perron root of X. Recall that X is irreducible if its directed graph is strongly connected or, equivalently, if it cannot be symmetrically permuted to a block upper triangular matrix having non-vacuous, square diagonal blocks. When X is irreducible, the Perron Frobenius theorem states that ρ(x) is a simple eigenvalue. In this case, we denote the corresponding unit right and left eigenvectors of X by x r (X) and x l (X), and refer to them as the unit right Perron vector and the unit left Perron vector, respectively. Let us now assume that A is nonnegative almost skew-symmetric and that S = ww t is irreducible. As S 0, we have that w must be a strictly positive vector and thus S is a strictly positive matrix. It follows that all L(A, α) (α (0, 1)) are also strictly positive (and thus irreducible) almost skew-symmetric matrices. As a consequence, φ(a,α) is a simple eigenvalue and thus the rightand left unitperron vectors of L(A, α) are well defined; in particular notice that x r (L(A, 1/)) = x l (L(A, 1/)) = x r (S) = x l (S) = w/ w. In addition, when S is irreducible, φ(a,α) is a differentiable function of α (0, 1). These facts underlie most of our statements in Sections 3 and 4. Example.1. This example illustrates the above definitions and that S being irreducible is indeed an assumption weaker than A being irreducible, even for nonnegative almost skew-symmetric matrices. The matrix

4 46 P.J. Psarrakos, M.J. Tsatsomeros / Linear Algebra and its Applications 360 (003) A = is a reducible nonnegative almost skew-symmetric matrix with K = and S = ww t, where w = It satisfies δ = 3andvar = 8/3. As S is irreducible, the main results herein apply to this matrix A. We proceed with a brief mention of some results from [11] that are relevant to our discussion. Notice that they apply to skew-symmetric matrices that are not necessarily nonnegative; in [11] an almost skew-symmetric matrix is by definition a matrix A whose symmetric part has rank one and δ > 0. Theorem.. Let A M n (R) be an almost skew-symmetric matrix and λ an eigenvalue of A. Then (Im λ) Re λ (δ Re λ) [var + Re λ(re λ δ)]. (.) Prompted by the above theorem, the shell of an almost skew-symmetric matrix A is defined in [11] as the curve in the complex plane given by { Γ = x + iy C : x,y R and ( )} var y = (δ x) + x δ. x By Theorem., Γ yields a localization of the spectrum of A, as specified by (.). The various possible configurations of the shell of an almost skew-symmetric matrix and how to achieve them via Levinger s transformation are illustrated in the next example. Example.3. We have taken a 5 5 almost skew-symmetric matrix A with δ =.6636 and variance var =.419, and found the almost skew-symmetric matrices B = L(A, 0.07) and C = L(A, 0.1) having variances var(b) = and var(c) = Of course, δ(b) = δ(c) = Their shells are shown in Fig. 1, where the eigenvalues of each matrix are marked with s. The shell Γ is connected and all the eigenvalues of A are located in the region between Γ and the imaginary axis. The shell Γ(C) consists of one bounded and one unbounded branch. The bounded branch surrounds a real eigenvalue of C and the unbounded branch isolates the rest of the spectrum. The shell of B is the transition in a continuous transformation from the connected shell to the shell with two branches.

5 P.J. Psarrakos, M.J. Tsatsomeros / Linear Algebra and its Applications 360 (003) Fig. 1. The shells Γ, Γ(B) and Γ(C). When A is almost skew-symmetric, applying Theorem. to L(A, α), whichis also almost skew-symmetric, the following can be shown [11]. Proposition.4. Let A M n (R) be an almost skew-symmetric matrix. Then for every α [0, 1/] such that δ > 4(1 α) var, the matrix L(A, α) has a real eigenvalue λ(a, α) δ + δ 4(1 α) var and n 1 complex eigenvalues whose real parts are not greater than δ δ 4(1 α) var. We conclude with an outline of our results to follow. In Section 3, we obtain bounds for the angles between the vectors w, x r (L(A, α)) and x l (L(A, α)) for appropriate values of α [0, 1/]. In Section 4, using the notion of q-numerical range and a result in [3], we construct an upper bound for the derivative of Levinger s function in (.1), which, in turn, yields an upper bound for φ(a,α). Finally, an illustrative example is presented in Section Right and left Perron vectors of L(A, α) Let A M n (R) be an irreducible nonnegative almost skew-symmetric matrix with symmetric part S = ww t. Then there exists an orthonormal basis of C n, {v 1,v,...,v n 1,w/ w }, where v 1,v,...,v n 1 Ker S and w/ w is the unit eigenvector of S corresponding to the eigenvalue δ = w t w. For any α [0, 1/], consider the unit right and left Perron vectors of L(A, α) and decompose them as x r (α) = y r (α) + z r (α) (3.1) and x l (α) = y l (α) + z l (α), (3.) where y r (α), y l (α) span {v 1,v,...,v n 1 } and z r (α), z l (α) span{w}.

6 48 P.J. Psarrakos, M.J. Tsatsomeros / Linear Algebra and its Applications 360 (003) Proposition 3.1. Let A M n (R) be an irreducible nonnegative almost skew-symmetric matrix. Then for every α [0, 1/], Levinger s function satisfies φ(a,α) = δ z r (α) = δ z l(α), where z r (α) and z l (α) are defined in (3.1) and (3.), respectively. Proof. It is enough to obtain that φ(a,α) = δ z r (α). The proof of the equality φ(a,α) = δ z l (α) is similar. Since the eigenvector x r(α) of L(A, α) in (3.1) is unit, it follows that Hence, φ(a,α) = x r (α) t L(A, α)x r (α). φ(a,α)= x r (α) t (S + (1 α)k)x r (α) = (y r (α) t + z r (α) t )S(y r (α) + z r (α)) + (1 α)x r (α) t Kx r (α) = z r (α) t Sz r (α) + (1 α)x r (α) t Kx r (α) = δ z r (α) + (1 α)x r(α) t Kx r (α). Note that φ(a,α) R and the matrix K is skew-symmetric. Thus, x r (α) t Kx r (α) = 0 and consequently, φ(a,α) = δ z r (α). Corollary 3.. Let A M n (R) be an irreducible nonnegative almost skew-symmetric matrix. Then for every α [0, 1/], the right and left Perron vectors of L(A, α) have the same orthogonal projection onto w, i.e., z r (α) = z l (α). Moreover, Levinger s function satisfies ( w t ) x r (α) ( w t ) x l (α) φ(a,α) = δ = δ. w w Proof. From Proposition 3.1, it follows that for every α [0, 1/], the vectors z r (α) and z l (α) have the same modulus. Also, since the Perron vectors x r (α) and x l (α) are nonnegative, and since z r (α) (resp., z l (α)) is the orthogonal projection of x r (α) (resp., x l (α)) onto the vector w, the claimed expression for φ(a,α) follows readily, as well as that cos (w, x r (α)) = cos( w, x l (α)) or, equivalently, ( w, x r (α)) = ( w, x l (α)). The latter equality and as z r (α) = z l (α), implies that z r (α) = z l (α).

7 P.J. Psarrakos, M.J. Tsatsomeros / Linear Algebra and its Applications 360 (003) Remark 3.3. Note that Proposition 3.1 and Corollary 3. hold for α (0, 1/] when the assumption of irreducibility of A is relaxed to irreducibility of the symmetric part of A. We will now apply the results in [11] in order to investigate the behavior of the Perron vectors of L(A, α). First, define the interval X A = max 0, 1 δ 16 var, 1 (3.3) and observe that α (0, 1/] lies in X A if and only if δ > 4(1 α) var. Notice also that since 0 / X A, if the symmetric part of A is irreducible, then for every α X A, L(A, α) is irreducible. Theorem 3.4. Let A M n (R) be a nonnegative almost skew-symmetric matrix with irreducible symmetric part. Then for every α X A, the cosine of the angle ( w, x r (α)) = ( w, x l (α)) is greater than or equal to the quantity 1 R A = + δ 4(1 α) var 4δ. Proof. We prove that cos( w, x r (α)) R A.Foranyα X A, by Proposition.4, φ(a,α) δ + δ 4(1 α) var. Hence, by Corollary 3. applied to L(A, α), it follows that δ(w t x r (α)) w or, equivalently, w t x r (α) w δ + δ 1 + Since the vector x r (α) is unit, it is clear that cos( w, x r (α)) 1 + δ 4(1 α) var 4δ δ 4(1 α) var 4δ. δ 4(1 α) var 4δ. Remark 3.5. These remarks refer to the above theorem. (i) The following interpretation of var is possible. For any α X A, define the angle ϑ A (α) [0, π/4] such that

8 50 P.J. Psarrakos, M.J. Tsatsomeros / Linear Algebra and its Applications 360 (003) cos ϑ A (α) = R A = + δ 4(1 α) var 4δ. (3.4) Consider now α varying in X A, starting from 1/ and taking values toward the left endpoint of X A.ThenthePerronvectorsx r (α) and x l (α) of L(A, α) lie in the cone K A (α) ={u R n : 0 (ŵ, u) ϑ A (α)}, which contains w. Notice that the smaller var is, the slower K A (α) dilates, and the slower x r (α) and x l (α) dilate away from w. (ii) For α = 1/, we have 1 1 = cos 0 = cos(ŵ, w) + δ 4δ = 1. Notice also that for every α X A, the angle ( w, x r (α)) = ( w, x l (α)) is not greater than π/4. Theorem 3.4 also allows us to estimate apriorithe angle between x r (α) and x l (α) and subsequently the Euclidean distance between these two vectors for any α X A. Theorem 3.6. Let A M n (R) be a nonnegative almost skew-symmetric matrix with irreducible symmetric part. Then for every α X A, cos( x r (α), δ x l (α)) 4(1 α) var δ. Proof. With ϑ A (α) as defined in (3.4) and for the angles ( x r (α), x l (α)), ( w, x r (α)) and ( w, x l (α)), we have ( x r (α), x l (α)) ( w, x r (α)) + ( w, x l (α)) ϑ A (α) π. Thus, cos( x r (α), x l (α)) cos(ϑ A (α)) 0, where cos(ϑ A (α)) = cos ϑ A (α) 1 δ = 4(1 α) var δ. Corollary 3.7. Let A M n (R) be a nonnegative almost skew-symmetric matrix with irreducible symmetric part. Then for every α X A, the Perron vectors x r (α) and x l (α) satisfy

9 P.J. Psarrakos, M.J. Tsatsomeros / Linear Algebra and its Applications 360 (003) x r (α) x l (α) 1 δ 4(1 α) var δ. Proof. Since the vectors x r (α) and x l (α) are unit, one can see that x r (α) x l (α) = (x r(α) t x l (α) t )(x r (α) x l (α)) = cos( x r (α), x l (α)). Hence, by the above theorem, x r (α) x l (α) δ 4(1 α) var δ. 4. Bounds for Levinger s function and its derivative Let A M n (R) be a nonnegative almost skew-symmetric matrix with irreducible symmetric part, and recall Levinger s function φ(a,α) = ρ(l(a, α)) = ρ((1 α)a + αa t ), α [0, 1/] and the interval X A (0, 1/] defined in (3.3). Proposition.4 applied to A, and since then λ(a, α) = φ(a,α), yields a lower bound on Levinger s function. In this section, we focus on upper bounds for φ(a,α) and for the rate of change of φ(a,α) as a function of α. The rate of change of Levinger s function has been studied systematically by Fiedler [3]. By the proof of [3, Theorem 1., p. 176], we have that for every α (0, 1/), 0 φ (A, α) = 1 x l (α) t (L(A, α) t L(A, α))x r (α) 1 α x l (α) t x r (α) = x l(α) t Kx r (α) cos( x r (α), x l (α)). (4.1) This expression for φ (A, α) will lead us to an upper bound for φ (A, α) that depends on var and δ; it is contained in Theorem 4.1. Subsequently, this upper bound will be used to obtain an upper bound for φ(a,α). To begin, by (4.1) and our Theorem 3.6, it follows that for every α X A \{1/}, 0 φ (A, α) ( xl (α) t Kx r (α) ), (4.) q(α) where q(α) = δ 4(1 α) var δ. (4.3)

10 5 P.J. Psarrakos, M.J. Tsatsomeros / Linear Algebra and its Applications 360 (003) At this point, it is necessary to introduce the notion of q-numerical range of an n n complex matrix N for a real q [0, 1], that is, F q (N) ={x Ny C : x,y C n,x x = y y = 1andx y = q} { x = N y } C : x,y C n \{0} and cos( x,y) = q. x y For q = 1,F q (N) coincides with the classical numerical range F(N) F 1 (N) ={x Nx C : x C n and x x = 1}. The q-numerical range F q (N) (q [0, 1]) is always a compact and convex subset of the complex plane. For 0 q<1,f q (N) has a nonempty interior, and for q = 1, the range F(N) is a line segment if and only if N is a normal matrix with colinear eigenvalues. In particular, a complex matrix N is Hermitian (resp., skew-hermitian) if and only if F(N) R (resp., F(N) ir). For the matrix A, it is obvious now that the quantity x l (α) t Kx r (α) in (4.) is positive and lies in the interval [0, + ) F cos( xr (α),x l (α)) (K). As a consequence, it will be necessary to estimate the length of the intersection of the positive half-axis [0, + ) with the cos( x r (α), x l (α))-numerical range of the real skew-symmetric matrix K. First observe that in general, for any pair 0 <α 1 <α < 1/, it follows that q(α 1 )<q(α ), where q(α) is defined in (4.3). Thus, by [8, Theorem.5], F q(α )(K) q(α ) q(α 1 ) F q(α 1 )(K) and consequently, [0, + ) F q(α )(K) q(α ) { [0, + ) Fq(α1 )(K) }. q(α 1 ) Similarly we obtain that for every α X A \{1/}, cos( x r (α), x l (α)) [0, + ) F cos( xr (α),x (K) l (α)) q(α) { [0, + ) F q(α) (K) }. Hence, by (4.) and Theorem 3.6, 0 φ (A, α) cos( x r (α), x l (α)) q (α) max { [0, + ) F q(α) (K) } q (α) max { [0, + ) F q(α) (K) }.

11 P.J. Psarrakos, M.J. Tsatsomeros / Linear Algebra and its Applications 360 (003) Theorem 4.1. Let A M n (R) be a nonnegative almost skew-symmetric matrix with irreducible symmetric part and let s 1 be the maximum singular value of the skewsymmetric matrix K. Then for every α X A \{1/}, 0 φ (A, α) 4s 1(1 α)δ var δ 4(1 α) var. Proof. The numerical range of K is F (K) =[ iβ,iβ], where the real number β is nonnegative [6]. Denote by D(λ, r) the closed disk centered at λ with radius r. By the results in [9], it is known that for any q [0, 1], F q (K) = (iγ,h) γ,h R D ( iqγ, (1 q )(h γ ) where the union is taken over all the pairs (iγ,h) = (x Kx, x (K K)x), x x = 1. Since K is skew-symmetric, there is a unitary matrix U such that K = iu diag{±s 1, ±s,...,±s m }U and { } K K = U diag s1,s 1,s,s,...,s m U, where s 1 s s m 0 are the singular values of K. (Note that every nonzero singular value appears an even number of times.) One can see that for the pair (iγ,h) = (0,s1 ), the radius of the disk ( ) D iqγ, (1 q )(h γ ) attains its maximum value, that is, r max (A, q) = s 1 1 q. Indeed, let y 1 and ŷ 1 be two orthonormal eigenvectors of K corresponding to the eigenvalues is 1 and is 1, respectively. Then the vector y 0 = (y 1 +ŷ 1 )/ isa unit eigenvector of matrix K K corresponding to s 1, and ), (y0 Ky 0,y0 (K K)y 0 ) ( y = 1 Ky 1 + ŷ 1 Kŷ ) 1,y0 (K K)y 0 = (0,s 1 ). Thus, for any α X A \{1/}, the set [0, + ] F q(α) (K) coincides with the interval

12 54 P.J. Psarrakos, M.J. Tsatsomeros / Linear Algebra and its Applications 360 (003) ] [ [0,s 1 1 q (α) = 0, s 1(1 α) ] var. δ Consequently, for any α X A \{1/}, we have 0 φ (A, α) q (α) s 1 (1 α) var, δ where q(α) is given in (4.3), or equivalently, 0 φ (A, α) 4s 1(1 α)δ var δ 4(1 α) var. Next we turn our attention to obtaining an upper bound for φ(a,α). The only upper bound we know so far is Levinger s result, namely, φ(a,α) δ.however, our bound for φ (A, α) can lead us to a better estimate for φ(a,α). Theorem 4.. Let A M n (R) be a nonnegative almost skew-symmetric matrix with irreducible symmetric part and let s 1 be the maximum singular value of the skewsymmetric matrix K. Then for every α 1,α X A \{1/} such that α 1 <α, 0 φ(a,α ) φ(a,α 1 ) s 1δ 4 var ln ( δ 4(1 α ) var δ 4(1 α 1 ) var Proof. From Theorem 4.1, for every α X A \{1/}, we get 0 φ (A, α) 4s 1(1 α)δ var δ 4(1 α) var. Integrating through the above inequality with respect to α in the interval (α 1, α ) X A \{1/}, and as φ(a,α)is a non-decreasing function in [0, 1/] (see [3,7]), we obtain the claimed inequality. Corollary 4.3. Let A M n (R) be a nonnegative almost skew-symmetric matrix with irreducible symmetric part and let s 1 be the maximum singular value of the skew-symmetric matrix K. If δ > 4var, then for every α [0, 1/), we have φ(a,α) ρ + s ( 1δ δ 4 var ln 4(1 α) ) var δ. 4var Moreover, for α = 1/, the following inequality obtains: δ ρ s ( 1δ 4 var ln δ ) δ. 4var Note that for values of α sufficiently close to 0, the upper bound for φ(a,α) in the above corollary is less than δ, and for α = 0, it coincides with ρ. ).

13 P.J. Psarrakos, M.J. Tsatsomeros / Linear Algebra and its Applications 360 (003) Illustrative example Consider the (irreducible) nonnegative matrix A = This matrix has irreducible symmetric part S = ww t, where w = ( /)[1, 1, 1, 1, 1, 1] t. Thus A is almost skew-symmetric. The spectral radius of A is ρ =.818, its variance is var = 0.53 and δ = 3. The maximum singular value of the skewsymmetric part of A, K = , is s 1 = We also have that XA = (0, 1/] and that for every α [0, 1/], 9 = δ >.1 4(1 α) var. If we choose α =1/3, then the Perron root of L(A, 1/3) is φ(a,1/3) = with corresponding unit Perron vectors x r (1/3) =[ , , , , , ] t and x l (1/3) =[ , , , , , ] t. By (4.1), we have φ (A, 1/3) = x l(1/3) t Kx r (1/3) = x l (1/3) t x r (1/3) Notice that q(1/3) = < = cos( x r (1/3), x l (1/3)), confirming Theorem 3.6. The upper bound of the derivative of Levinger s function in Theorem 4.1 is 4s 1 (1 /3)δ var δ 4(1 /3) var = > = φ (A, 1/3).

14 56 P.J. Psarrakos, M.J. Tsatsomeros / Linear Algebra and its Applications 360 (003) Fig.. The lower and upper bound for φ(a,α). In Fig., we illustrate the upper bound for φ(a,α) in Corollary 4.3 and the lower bound in Proposition.4. The Perron roots φ(a,α) for α = 0, 0.1, 0., 0.3, 0.4, 0.5 are marked with o s. Notice that since the variance var = 0.53 is relatively small, the lower bound in Proposition.4 is evidently quite close to Levinger s function φ(a,α). Finally, we remark that Levinger s function and our bounds for it have the same qualitative behavior in X A, that is, they are all increasing and concave functions, and their derivatives have a zero at α = 1/. References [1] A. Berman, R.J. Plemmons, Nonnegative Matrices in the Mathematical Sciences, SIAM, Philadelphia, PA, [] M. Fiedler, Geometry of the numerical range of matrices, Linear Algebra Appl. 37 (1981) [3] M. Fiedler, Numerical range of matrices and Levinger s theorem, Linear Algebra Appl. 0 (1995) [4] S. Friedland, Eigenvalues of almost skew-symmetric matrices and tournament matrices, Combinatorial and graph-theoretical problems in linear algebra, in: R.A. Brualdi, S. Friedland, V. Klee (Eds.), IMA vol. Math. Appl., vol. 50, Springer, New York, 1993, pp [5] S. Friedland, M. Katz, On the maximal spectral radius of even tournament matrices and the spectrum of almost skew-symmetric compact operators, Linear Algebra Appl. 08/09 (1994) [6] R.A. Horn, C.R. Johnson, Topics in Matrix Analysis, Cambridge University Press, Cambridge, [7] B.W. Levinger, An inequality for nonnegative matrices, Notices Am. Math. Soc. 17 (1970) 60.

15 P.J. Psarrakos, M.J. Tsatsomeros / Linear Algebra and its Applications 360 (003) [8] C.-K. Li, P. Metha,, L. Rodman, A generalized numerical range: the range of a constrained sesquilinear form, Linear and Multilinear Algebra 37 (1998) [9] C.-K. Li, H. Nakazato, Some results on the q-numerical range, Linear and Multilinear Algebra 43 (1998) [10] J. Maroulas, P.J. Psarrakos, M.J. Tsatsomeros, Perron-Frobenius type results on the numerical range, Linear Algebra Appl. 348 (00) [11] J. McDonald, P. Psarrakos, M. Tsatsomeros, Almost skew-symmetric matrices, Rocky Mountain J. Math., to appear.