Bounds for Levinger s function of nonnegative almost skew-symmetric matrices

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1 Linear Algebra and its Applications Bounds for Levinger s function of nonnegative almost skew-symmetric matrices Panayiotis J. Psarrakos a, Michael J. Tsatsomeros b, a Department of Mathematics, National Technical University, Zografou Campus, Athens 15780, Greece b Mathematics Department, Washington State University, Pullman, WA , USA Received 15 February 005; accepted 0 December 005 Available online 9 February 006 Submitted by S. Kirkland Abstract The analysis of the Perron eigenspace of a nonnegative matrix A whose symmetric part has rank one is continued. Improved bounds for the Perron root of Levinger s transformation 1 αa + αa t α [0, 1] and its derivative are obtained. The relative geometry of the corresponding left and right Perron vectors is examined. The results are applied to tournament matrices to obtain a comparison result for their spectral radii. 006 Elsevier Inc. All rights reserved. AMS classification: 15A18; 15A4; 15A60; 05C0 Keywords: Perron root; Perron vector; Levinger s function; Tournament 1. Introduction Our main goal is to study the spectrum and, in particular, the spectral radius of an entrywise nonnegative matrix whose symmetric part has rank one. We refer to such a matrix A as nonnegative almost skew-symmetric. Our motivation lies in the fundamental nature of this problem within the realm of nonnegative matrix theory, as well as the relation and applications of such Research supported by a grant of the EPEAEK project PYTHAGORAS II. The project is co-funded by the European Social Fund 75% and Greek National Resources 5%. Corresponding author. addresses: ppsarr@math.ntua.gr P.J. Psarrakos, tsat@math.wsu.edu M.J. Tsatsomeros /$ - see front matter 006 Elsevier Inc. All rights reserved. doi: /j.laa

2 760 P.J. Psarrakos, M.J. Tsatsomeros / Linear Algebra and its Applications matrices to the theory of tournaments see [6,8,9,11]. A key role in these papers was played by Levinger s transformation, i.e., 1 αa + αa t α [0, 1]. This transformation has also proven useful in answering fundamental questions about the geometry of the numerical range of a general or nonnegative matrix [10]. In Sections 3 and 4, we continue the analysis of the Perron eigenspace of a nonnegative almost skew-symmetric matrix started in [11] and obtain new bounds for the Perron root of Levinger s transformation and its derivative. These concepts were first systematically studied by Fiedler [4]. Section 5 comprises an illustrative example. The association to tournament matrices, relevant definitions and some observations relating to their spectral radii and the Brualdi Li conjecture are presented in Section 6.. Preliminaries Let x,y R n be two real vectors. We call x a unit vector if its Euclidean norm is x = 1. The angle between x and y is defined by x,y = cos 1 x t y [0, π]. x y Consider an n n real matrix A and denote its spectrum by σ A and its spectral radius by ρa = max{ λ :λ σ A}. For every real square matrix A, we write A = SA + KA, where SA = A + At and KA = A At are the real symmetric part and the real skew-symmetric part of A, respectively. Given any square matrix A, we also consider Levinger s transformation, LA, α = 1 αa + αa t, α [0, 1], as well as Levinger s function, φa,α = ρla, α = ρ 1 αa + αa t, α [0, 1]. 1 Note that as Levinger s function is symmetric about α = 1/, without loss of generality, we will hereon restrict our attention to α [0, 1/]. One can also readily see that for every α [0, 1/], LA, α = SA + 1 αka. Given an entrywise nonnegative square matrix A, by the Perron Frobenius theory, we know that ρa is an eigenvalue of A, to which we shall refer as the Perron root of A. Corresponding to ρa are nonnegative unit right and left eigenvectors referred to as Perron vectors and denoted, respectively, by x r A and x l A. Continuing to assume that A is an n n nonnegative matrix, we of course have that SA is also nonnegative. Let us further assume that rank SA = 1; i.e., there exists a nonzero nonnegative vector w R n such that SA = ww t. This means that σ SA consists of the eigenvalue 0 with multiplicity n 1 and a simple positive eigenvalue δa = w t w. We refer to a matrix A having all of the features in this paragraph as a nonnegative almost skew-symmetric matrix. We will from now on presume that the notation associated with such a matrix A is readily recalled. Note that when A is an irreducible nonnegative almost skew-symmetric matrix, all the matrices LA, α α [0, 1/] are also irreducible nonnegative almost skew-symmetric, having

3 P.J. Psarrakos, M.J. Tsatsomeros / Linear Algebra and its Applications the same symmetric part, SA. Following the terminology in [9,11], we define the variance of A by vara = KAw w = wt KA t KAw w t, w and observe that the variance of LA, α is varla, α = 1 α vara. Furthermore, φa,0 = ρa, φa, 1/ = ρsa = δa and x r LA, 1/ = x l LA, 1/ = x r SA = x l SA = w/ w. Next, we summarize some results in [9,11] needed in our discussion. Theorem 1. Consider an n n nonnegative almost skew-symmetric matrix A with irreducible symmetric part. Then Levinger s function φa,α defined in 1 and the corresponding unit Perron vectors x r α = x r LA, α and x l α = x l LA, α satisfy the following: a For every α [0, 1/] such that δa > 41 α vara, φa,α δa + δa 41 α vara. b For every α 0, 1/] such that δa > 41 α vara, the cosine of the angle w, x r α = w, x l α is greater than or equal to the quantity 1 + δa 41 α vara 4δA. c If we let s 1 = KA = ρka, then for every α 0, 1/ such that δa > 41 α vara, 0 φ A, α 4s 11 αδa vara δa 41 α vara. d If δa > 4varA, then for every α [0, 1/, φa,α ρa + s 1δA δa 4 vara ln 41 α vara δa. 4varA Finally in this section, we recall a useful inequality of Rojo et al. [1]. Theorem. For any eigenvalue λ of an n n real matrix A, tracea Re λ n n 1 SA n AAt A t A F 1 A F tracea, n where F denotes the Frobenius norm.

4 76 P.J. Psarrakos, M.J. Tsatsomeros / Linear Algebra and its Applications Perron roots and Perron vectors Let A be an n n nonnegative almost skew-symmetric matrix with symmetric part SA = ww t, skew-symmetric part KA and variance vara. The quantity AA t A t A F is known as the Frobenius distance to normality of A. Next, we obtain a formula for this distance that is necessary for the remainder and of independent interest. Proposition 3. The distance to normality of an n n nonnegative almost skew-symmetric matrix A is AA t A t A F = δa vara. Proof. Let y = w/ w R n be the unit eigenvector of SA = ww t corresponding to δa. Then there is an n n real unitary matrix V, whose first column is y, such that V t SAV = diag{δa, 0,...,0}. Moreover, since V t KAV is real skew-symmetric, it follows [ ] V t 0 u t KAV =, u K 1 where K 1 is n 1 n 1 real skew-symmetric and u R n 1. Observe now that AA t A t A = SA + KASA KA SA KASA + KA = KASA SAKA, and since the Frobenius norm is invariant under unitary similarities, AA t A t A F = 4 V t KAV V t SAV V t SAV V t KAV F [ ][ ] [ ][ ] = 4 0 u t δa 0 δa 0 0 u t u K u K 1 F [ ] [ ] = δau t + δau F [ ] = 4δA 0 u t u 0 F = 8δA u. Denoting by e 1 the first standard basis vector in R n,wehave [ ] u 0 [ ] = = 0 u u V t e u K 1 1 [ ] = 0 u V t V t Ve u K 1 1 = KAy = vara. As a consequence, AA t A t A F = 8δA vara.

5 P.J. Psarrakos, M.J. Tsatsomeros / Linear Algebra and its Applications Corollary 4. An n n nonnegative almost skew-symmetric matrix A is normal if and only if vara = 0. The Frobenius norm of A may be written as A F = SA + KA F = SA F + KA F, and one can verify that A F = δa + KA F. Then Theorem yields the following result. Theorem 5. The spectral radius of an n n nonnegative almost skew-symmetric matrix A satisfies ρa δa 1 n + n 1 n 1 vara n n 3 δa + KA. F Proof. By Proposition 3,wehave AA t A t A F = 8δA vara. Since A F = δa + KA F and tracea = tracesa = δa = SA = w, the proof is complete by Theorem. Notice that if vara 0, the upper bound of the spectral radius ρa in approaches δa, which is the maximum possible value for ρa. Furthermore, for any α [0, 1/], LA, α F = δa + 1 α KA F. Corollary 6. Let A be an n n nonnegative almost skew-symmetric matrix. Then for every α [0, 1/], φa,α δa 1 n + n 1 n 1 1 α vara n n 3 δa + 1 α KA. F Equality holds when α = 1/. Next is a comparison result for the spectral radii of nonnegative almost skew-symmetric matrices to be cited in Section 6. Proposition 7. Let A and B be n n nonnegative almost skew-symmetric matrices with variances vara and varb, and assume that δa = δb.ifδb > 4varB and if 1 n + n 1 n 1 vara varB δb n n 3 A <, 3 F then ρa < ρb. Proof. It follows readily from Theorems 1a and 5.

6 764 P.J. Psarrakos, M.J. Tsatsomeros / Linear Algebra and its Applications By the results in [11], we know that if the symmetric part of A is irreducible, then for every α [0, 1/], the right and left unit Perron vectors x r α and x l α of LA, α have the same orthogonal projection onto the vector w see also Theorem 1b and satisfy w t x r α w t x l α φa,α = δa = δa. 4 w w Note that all the bounds in Proposition 8 and in Corollaries 9 and 10 below approach 1 as α 1/. Proposition 8. Let A be an n n nonnegative almost skew-symmetric matrix. If SA is irreducible, then for every α [0, 1/], the cosine of the angle w, x r α = w, x l α is less than or equal to 1 n + n 1 n 1 1 α vara n n 3 δa + 1 α KA. F Proof. By Corollary 6, for every α [0, 1/], φa,α δa 1 n + n 1 n 1 1 α vara n n 3 δa + 1 α KA. F Hence, by 4, w t x r α 1 w n + n 1 n 1 1 α vara n n 3δA + 1 α KA F. The Perron vector x r α is a unit vector and the proof is complete. For a nonnegative matrix A, by results found, e.g., in [5,10], LA, α F n max{ x Ax :x C n, x = 1} = nδa. Thus, the above proposition and Theorem 1b imply the following corollaries. Corollary 9. Let A be an n n nonnegative almost skew-symmetric matrix. If SA is irreducible, then for every α [0, 1/], the cosine of the angle w, x r α = w, x l α is less than or equal to 1 n + n 1 n 1 1 α4 vara n n 6nδA. Corollary 10. Let A be an n n nonnegative almost skew-symmetric matrix. If SA is irreducible, then for every α [0, 1/] such that δa > 41 α vara, the cosine of the angle w, x r α = w, x l α lies in the interval α vara n 1 1 δa, n + n 11 α4 vara n 6n δa.

7 P.J. Psarrakos, M.J. Tsatsomeros / Linear Algebra and its Applications Bounds for Levinger s function and its derivative Let A be an n n nonnegative almost skew-symmetric matrix. Consider Levinger s function φa,α = ρla, α = ρ1 αa + αa t α [0, 1/] and the open interval X A = max 0, 1 δa 16 vara, 1. 5 Notice that α 0, 1/ lies in X A if and only if δa > 41 α vara. We shall first obtain a new upper bound for φ A, α that is tighter than the one in Theorem 1c. Note that, by the proof of [4, Theorem 1.], for every α [0, 1/, 0 φ A, α = 1 x l α t LA, α t LA, αx r α 1 α x l α t x r α = x lα t KAx r α cos x l α, x r α. 6 Theorem 11. Let A be an n n nonnegative almost skew-symmetric matrix with irreducible symmetric part. Then for every α X A, φ A, α = φa,α 1 1 α cos x r α, x l α 1 δa δa δa 41 α vara 1 α. 7 δa 41 α vara Proof. By 6 and the definitions, the following equalities ensue: φ A, α = x lα t KAx r α cos x r α, x l α = 1 α = 1 α x l α t LA, α SAx r α cos x r α, x l α φa,α x lα t SAx r α cos x r α, x l α Letting zα denote the orthogonal projection of x r α and x l α onto w,by[11, Proposition 3.1], we have x l α t SAx r α = zα t SAzα = δazα t zα = δa zα t = φa,α,.

8 766 P.J. Psarrakos, M.J. Tsatsomeros / Linear Algebra and its Applications and hence, φ A, α = φa,α 1 α φa,α cos x r α, x l α By the above equality, the fact that φa,α δa, and by [11, Theorem 3.6], the proof is complete. It is worth noting that the upper bound in 7 for the derivative of Levinger s function is independent of s 1 = KA. Furthermore, since s 1 KAw = vara, w by straightforward computations, one can see that for every α X A, s 1 δa δa 41 α vara δa 41 α vara vara > 1 α. vara As a consequence, the upper bound in the previous theorem is an improvement over the upper bound provided in Theorem 1c for all α X A. Integrating through 7 with respect to α in an interval α 1,α X A, we obtain the following result. Theorem 1. Let A be an n n nonnegative almost skew-symmetric matrix with irreducible symmetric part. Then for every α 1,α X A with α 1 <α, φa,α φa,α 1 δa ln 1 α δa δa 41 α 1 vara 1 α 1 δa. δa 41 α vara Moreover, if X A = 0, 1/, i.e., if δa 4varA, then for every α 0, 1/], φa,α ρa + δa ln 1 α δa δa 4varA δa. δa 41 α vara Theorem 13. Let A be an n n nonnegative almost skew-symmetric matrix with irreducible symmetric part. Then for every α 1 <α in X A, φa,α 1 α δa δa 41 α 1 vara φa,α 1 1 α 1 δa. δa 41 α vara Proof. By Theorem 11,wehave 1 αφ A, α 1 = φa,α cos x r α, x l α 1, or equivalently, 1 cos x r α, x l α = φa,α + 1 αφ A, α. φa,α.

9 P.J. Psarrakos, M.J. Tsatsomeros / Linear Algebra and its Applications Furthermore, by [11, Theorem 3.6], δa 41 α vara δa cos x r α, x l α φa,α = φa,α + 1 αφ A, α, and hence, + 1 α φ A, α φa,α δa δa 41 α vara. Thus for every α X A, 0 φ A, α φa,α 1 α + δa 1 α δa 41 α vara. Integrating through the above inequality with respect to α in the interval α 1,α X A obtains ln φa,α φa,α 1 ln 1 α δa δa 41 α 1 vara 1 α 1 δa δa 41 α vara and consequently, φa,α 1 α δa δa 41 α 1 vara φa,α 1 1 α 1 δa. δa 41 α vara The proof is complete. If δa > 4varA, then for α = α 0, 1/] and α 1 0, 1 α δa δa 4varA φa,α ρa δa. 8 δa 41 α vara Furthermore, by 4, the cosine of the angle w, x r α = w, x l α is less than or equal to ρa1 α δa δa 4varA δa δa. 9 δa 41 α vara Letting now α 1/ and α 1 = α X A, it follows φa,α vara δa δa 41 α vara. Note also that by straightforward computations, one can see that the latter lower bound of φa,α is exactly the same as the bound provided in Theorem 1a.

10 768 P.J. Psarrakos, M.J. Tsatsomeros / Linear Algebra and its Applications An illustration The 6 6 irreducible nonnegative matrix A = is almost skew-symmetric and has irreducible symmetric part SA = 11 t, where 1 R n denotes the all ones vector. The Perron root of A is ρa = , its variance is vara = , and δa = 6. The skew-symmetric part of A is KA = and the Frobenius norms of the matrices A, SA and KA are A F = 7.77, SA F = 6 and KA F = The condition δa > 4varA is satisfied. In Fig. 1, our bounds for Levinger s function φa,αare illustrated. The Perron roots φa,α = ρla, α for α = 0, 0.05, 0.1,...,0.5 are plotted by +. The curve a is the lower bound in Theorem 1a, the curve b is the upper bound in Corollary 6, the curve c is the upper bound in the second part of Theorem 1, and the curve d is the upper bound in c d 5.95 b a Fig. 1. The bounds for φa,α.

11 P.J. Psarrakos, M.J. Tsatsomeros / Linear Algebra and its Applications D 1 B A Fig.. The bounds for the cosine of w, x r α = w, x l α. Our bounds for the cosine of the angle w, x r α = w, x l α are verified in Fig., where the values of the cosine for α = 0, 0.05, 0.1,...,0.5 are also plotted by +. The curve A is the lower bound in Theorem 1b, the curve B is the upper bound in Proposition 8, and the curve D is the upper bound in 9. Notice that these three bounds have exactly the same behaviour as the bounds provided by the curves a, b and d in Fig. 1, respectively. 6. Tournament matrices One of the most intriguing problems in combinatorial matrix theory regards a conjecture posed by Brualdi and Li []. It states that among all tournament matrices of a given even order, the maximal spectral radius is attained by the Brualdi Li matrix, defined below. This conjecture has been confirmed for small sizes, and there is supporting evidence for its validity asymptotically as the order grows large; see [3,7]. In this section, we suggest an alternative approach and report some related progress. We provide a comparison result for spectral radii and show that the spectral radius of the Brualdi Li matrix is maximum among n n tournament matrices whose score variance exceeds a certain function of n. We begin by reviewing some basic facts about tournament matrices. Recall that an n n tournament matrix T is a 0, 1-matrix such that T + T t = J I n, where J = 11 t. Recall that 1 denotes an all ones vector. For odd n, T is called regular if all its row sums equal n 1/, i.e., T 1 =[n 1/]1. For even n, T cannot have all row sums equal; in this case, T is called almost regular if half its row sums equal n/ and the others equal n /. To make the connection to our previous sections, note that if T is an n n tournament matrix, then A = T + 1/I n is nonnegative almost skew-symmetric with SA = 1/J. The score variance of T is defined by

12 770 P.J. Psarrakos, M.J. Tsatsomeros / Linear Algebra and its Applications svt = 1 n 1 n T 1 1. Consequently, svt = 1 n KT1 = 1 n KA1 = vara. Moreover, δa = n/, ρt = ρa 1/ and A nn 1 F = + n 4 = n n. 4 Clearly, the score variance of a regular tournament equals 0 and hence, by Corollary 4, all regular tournaments are normal matrices and the score variance of an almost regular tournament is 1/4. In general, svt n 1/1 with equality holding when T is triangular. It has been shown that for sufficiently large even n, the tournament matrix attaining the maximum Perron root among all n n tournament matrices must be almost regular see [7, Theorem 3]. In addition, as shown in [3], among the Perron roots of all almost regular n n tournament matrices of the form [ T T BT = t ] T t, + I n/ T where T is itself a tournament matrix of order n/, the maximum is attained when T is the strictly upper triangular matrix U all of whose entries above the main diagonal are equal to 1. We then refer to B = BU as the Brualdi Li matrix. The Brualdi Li conjecture states that ρb maximizes the Perron root among all n n tournament matrices of even order n. Consider now two n n tournament matrices T 1 and T, as well as the corresponding almost skew-symmetric matrices A 1 = T 1 + 1/I n and A = T + 1/I n. By the above discussion, inequality 3 in Proposition 7 can be re-written as n 1 n 1 8varA 1 <n + n 6n 3 16 vara, or equivalently, vara 1 > 6n 3 n n + 8varA n n 16n 1 16 vara. Thus, the following comparison result for the spectral radii of tournaments is valid as a consequence of Proposition 7. Theorem 14. Let T 1 and T be two n n tournament matrices with score variances svt 1 and svt such that n > 16 svt.if svt 1 > 6n 3 n n + 8svT n n 16n 1 16 svt, 10 then ρt 1 <ρt. Next, notice that the function hv = 6n 3 16n 1 n n + 8v n n 16v, 0 v n 16

13 P.J. Psarrakos, M.J. Tsatsomeros / Linear Algebra and its Applications that appears on the right hand side of 10 is increasing. In fact, h v = 6n n > 4, 0 <v< n n 1 n 16v 16, and hv satisfies n h0 = 0 and h = 16 3nn 13n 4. 3n 1 For even n, the smallest possible score variance of an n n tournament matrix is 1/4 and almost regular tournament matrices attain this value. Thus, Proposition 3 implies that almost regular tournaments attain the minimum distance to normality, n /, among all n n tournament matrices. Furthermore, the smallest score variance among the rest of n n tournament matrices that is, excluding the almost regular ones is 1/4 + /n. As a consequence, keeping in mind that the sequence 6n 3 16n 1 n n + n n 4, n =, 3,... is increasing and converges to 3/, Theorem 14 yields the following result related to the Brualdi Li conjecture see also [7, Theorem 3]. Corollary 15. Let n 4 be even, and let T 0 be an n n almost regular tournament matrix. For every n n tournament matrix T with svt > 6n 3 n n + n n 16n 1 4, we have ρt<ρt 0. Moreover, if svt > 3/, then ρt<ρt 0. Next, we mention that our results can be used to prove another result related to the Brualdi Li conjecture, which is already known; see [6, Corollary 1.4 and preceding commentary]. Corollary 16 [6, Kirkland]. Let T 0 be an n nneven almost regular tournament matrix. Then, ρt 0 n + n 4 4 and lim ρt 0 n 1 = 0; n that is, as the order n increases, the spectral radius of an almost regular tournament of order n approaches the maximum possible value among the spectral radii of all tournament matrices of order n. Proof. It is known that the real part of every eigenvalue of an n n tournament matrix and thus its spectral radius is bounded above by n 1/ [1]. Hence, by the discussion so far and in conjunction with Theorem 1a applied to T 0 + 1/I n,wehave n + n 4 4 ρt 0 n 1.

14 77 P.J. Psarrakos, M.J. Tsatsomeros / Linear Algebra and its Applications However, it is easy to verify that lim n completing the proof. n + n 4 n 1 4 = 0, In conclusion, we note that our treatment of tournament matrices does not depend strongly on their special 0, 1-structure; rather, it is based on analytic techniques that suggest a new possible approach toward the Brualdi Li conjecture. In particular, it is of interest and would possibly lead to a positive resolution of the conjecture if one were able to reach the conclusion of Theorem 14 under a relaxed upper bound for the score variance of T 1 in 10. References [1] A. Brauer, I.C. Gentry, On the characteristic roots of tournament matrices, Bull. Amer. Math. Soc [] R.A. Brualdi, Q. Li, Problem 31, Discrete Math [3] C. Eschenbach, F. Hall, R. Hemasinha, S.J. Kirkland, Z. Li, B.L. Shader, J.L. Stuart, J.R. Weaver, On almost regular tournament matrices, Linear Algebra Appl [4] M. Fiedler, Numerical range of matrices and Levinger s theorem, Linear Algebra Appl [5] R.A. Horn, C.R. Johnson, Topics in Matrix Analysis, Cambridge University Press, Cambridge, [6] S. Kirkland, Hypertournament matrices, score vectors and eigenvalues, Linear and Multilinear Algebra [7] S. Kirkland, Perron vector bounds for a tournament matrix with applications to a conjecture of Brualdi and Li, Linear Algebra Appl [8] S. Kirkland, P. Psarrakos, M. Tsatsomeros, On the location of the spectrum of hypertournament matrices, Linear Algebra Appl [9] J. McDonald, P. Psarrakos, M. Tsatsomeros, Almost skew-symmetric matrices, Rocky Mountain Journal of Mathematics [10] J. Maroulas, P. Psarrakos, M. Tsatsomeros, Perron Frobenius type results on the numerical range, Linear Algebra Appl [11] P. Psarrakos, M. Tsatsomeros, The Perron eigenspace of nonnegative almost skew-symmetric and Levinger s transformation, Linear Algebra Appl [1] O. Rojo, R. Soto, H. Rojo, New eigenvalues estimates for complex matrices, Comput. Math. Appl