Linear Algebra and its Applications 47 7 9 9 www.elsevier.co/locate/laa A new type of lower bound for the largest eigenvalue of a syetric atrix Piet Van Mieghe Delft University of Technology, P.O. Box 356, 6 AJ Delft, The Netherlands Received Septeber 6; accepted 8 June 7 Available online 4 August 7 Subitted by S. Friedland Abstract Let A denote a syetric atrix. We present an order expansion 4 based on Lagrange series that allows us to iprove the classical bound i= j= a ij λ ax A. 7 Elsevier Inc. All rights reserved. AMS classification: Priary 5A4; Secondary 3B, 4A58 Keywords: Largest eigenvalue; Syetrical atrix; Spectra of graphs; Lagrange series; Perturbation theory. Introduction Let λ in A λ λ λ ax A denote the ordered, real eigenvalues of a syetric atrix A. The largest eigenvalue can be expressed [4, p. 549] as λ ax A = ax x/= x T Ax x T x, where xt Ax is called the Rayleigh quotient. The axiu in is only attained if x is the x T x eigenvector belonging to λ ax A. Hence, for any other vector y that is not the corresponding eigenvector, it holds that y T Ay y T y λ axa E-ail address: P.VanMieghe@ewi.tudelft.nl 4-3795/$ - see front atter 7 Elsevier Inc. All rights reserved. doi:.6/j.laa.7.6.7
P. Van Mieghe / Linear Algebra and its Applications 47 7 9 9 fro which the coonly used lower bound, u T Au = a ij λ ax A i= j= follows for the choice y = u, where u = [ ] T is the all-one vector. The ain result of this article is Theore.. Consider a syetric atrix A and define T = ax j a jj + i=;i/=j a ij. For any real nuber t T and λ = t, the largest eigenvalue of A can be bounded fro below by N + N 3 N N + N 3 3 where N k = u T A k u = i= j= A k ij and N =. λ + Ot 4 λ ax A, 3 Although the theore is stated with order ter Ot 4, the ethod of Appendix B allows us to sharpen the classical bound to any desired order Ot j, where j is a positive integer. If the λ ter and the order ter are ignored, we find the classical bound. In the theory on the spectra of graphs, nuerous lower and upper bounds for the largest eigenvalue λ ax A of the adjacency atrix A of a graph G exist see e.g. [6, Appendix B]. In a graph, N k is the total nuber of walks of length k [6, Appendix B] and the degree of node k is d k A = j= A kj. Then, the first few N k are N = L A, where L A is the nuber of links in G and N = D A = dk A. For graphs, we show in Section 3 that 3 can be rephrased as L A + N 3 L AD A + 4L3 A 3 λ + ON λ ax A 4 for any N and where λ = N. Equality in the classical bound is attained in regular graphs where each node has the sae degree r and where λ ax A = L A = r. Thus, for regular graphs, the coefficient of the λ ter in 4 is precisely zero. In Section, Theore. is proved. Section 3 discusses the applications to graphs, while Section 4 revisits Theore. fro the viewpoint of perturbation theory.. Proof of Theore. The ingredients of the proof of Theore. rely on the possibility to copute the largest eigenvalue λ ax A t and the sallest eigenvalue λ in A t of the syetric atrix [ ] A t.u A t = t.u T, where t R and on Lea A. that yields λ ax A t + λ in A t λ ax A. 5 The sequel is devoted to the coputation of λ ax A t and λ in A t. Lea A. restricts the validity of the analysis to syetric atrices.
P. Van Mieghe / Linear Algebra and its Applications 47 7 9 9 The characteristic polynoial of A t is [ ] A λi t.u deta t λi + + = t.u T. λ The general relation 5 gives deta t λi = deta λi det λ t u T A λi u. For any atrix X, the su of all its eleents is s X = u T Xu = n i= nj= x ij. Let us denote by s λ the su of all eleents of the resolvent A λi, then deta t λi = λ + t s λ deta λi. 6 An explicit expression for s λ is given in 4. Consider the expansion of the resolvent of A, A λi = I A = A k I λ λ λ λ k = Aλ A + + λ λ + such that s λ = λ i= j= k= A k ij λ k = λ k= k= λ k A k ij = λ i= j= Introduced into the equation λ + s λ t =, gives λ + s λ t = λ t N k + λ λ k = 8 or, k= N k λ k. λ =±t N k + λ k. 9 If t is large and fixed, 9 reveals that the first order expression, λ =±t is accurate up to O. Gerschgorin s Theore [7, p. 7] shows that each eigenvalue of the syetric atrix A lies in at least one of the j intervals a jj i=;i/=j a ij,a jj + i=;i/=j a ij. Hence, if t>t= ax j a jj + i=;i/=j a ij, then relation 9 further shows that the largest eigenvalues of A t in absolute value are the roots of λ + s λ t =. This observation leads to a lower bound for t T... Lagrange series for the zero of 8 We solve the equation f λ = λ + s λ t λ t λ + N + N λ = by Langrange expansion first up to Ot while the general ethod is outlined in Appendix B. The zero ζλ of f λ around λ = t can be written as a Lagrange series [3, II, pp. 88]. Since f λ is known up to order Ot, only three ters in the general Lagrange series are needed, λ 3 7 The sae result can be obtained by two iterations in Newton-Raphson s ethod.
P. Van Mieghe / Linear Algebra and its Applications 47 7 9 9 ζλ λ f λ f λ f λ f λ f λ f λ to guarantee that the zero is also accurate up to order Ot. Indeed, f λ = N N t + Ot, f λ = + N t + Ot, f λ = t + Ot and f λ = Ot. After substitution in andsoereorganization,theextreeeigenvalues of A t are, accurate up to order Ot for large t, λ ax A t = t + N + N 3N 8 t + Ot and λ in A t = t + N N 3N 8 t + Ot, where the last expression is obtained analogously fro for λ = t. The ethod presented in Appendix B allows us to copute ζλ to any desired order in t, although the aount of coputations rapidly becoes ipressive. Coputed up to Ot 3, we find ζλ = λ + N + N 3N 8 λ + N 3 N N + N 3 3 λ + Ot 3, 3 where λ ax A t and λ in A t follow for λ = t and λ = t, respectively. Finally, our Theore., in particular the bound 3, is proved by cobining the bound 5 and equation 3. 3. Application to the spectra of graphs Since A t is not an adjacency atrix for t /=, we consider the adjacency atrix of the G- connected star topology with N nodes, [ ] A J A starg = N, J N O N N where J is the all-one atrix and A is the adjacency atrix of an arbitrary graph G that connects nodes. Each of those nodes is connected to each of the N other nodes in the topology called starg. We note that the bi-partite structure of A starg is crucial. Siilarly as above, we find deta starg λi = deta λi N λ N λ + s λ N, where t = N. Hence, by odifying the size of the atrix A starg, the zeros of the sae function f λ = λ + s λ N are the axiu and iniu eigenvalue of A starg. Moreover, since the largest eigenvalue of A for any graph is saller than the axiu degree d ax A, a tighter bound for N copared to t is found, N. The general result is then given in 4.
P. Van Mieghe / Linear Algebra and its Applications 47 7 9 9 3 Exaple. The spectru of a -fully eshed star topology where A = J I can be coputed exactly as λ ax star = N + +, λ in star = N + + and with an eigenvalue at with ultiplicity and at with ultiplicity N. Coparing 3 with the exact result of a -fully eshed star topology, λ ax star = N + + = N + + + ON 3/ N shows, indeed, that 3 is correct, since N =, N = and N 3 = 3 for the coplete graph K. 4. Perturbation theory Consider the syetrix atrix [ ] O u B = u T that represents the adjacency atrix of the bi-partite graph K, or the star topology, a central node that connects other, not interconnected nodes. The eigenvalues of B are well-known:, and with ultiplicity. The corresponding eigenvectors to the eigenvalue and are v =[u ] T and w =[u ] T, respectively. Hence, apart fro the zero eigenvalues, the eigenvalues of tb are precisely λ =±t. Further, we can write A t = tb + A = t B + t A which iplies that the eigenvalues of A t are equal to those of B + t A ultiplied by t. Since we known the eigenvalues of B exactly, and if t is sufficiently large, perturbation theory [,7] can be applied. Since B + za is analytic in z, real syetric on the real axis, all eigenvalues of B + za are analytic functions of z in the neighborhood of the real axis Iz =. Hence, there exists a real nuber R>, for which B + za has two, siple eigenvalues λ + z and λ z with Taylor expansion around and, λ + z = + λ z = + α k z k β k z k z <R, z <R,
4 P. Van Mieghe / Linear Algebra and its Applications 47 7 9 9 where all coefficients α k and β k are real. Perturbation theory [7, p. 69] gives explicitly the first order coefficients as Thus, α = vt Av v T v = ut Au and β = wt Aw w T w = ut Au. α = β = N. For sufficiently sall z, λ + z and λ z are the axiu and iniu eigenvalue of B + za. Hence, with 5, we obtain λ ax A t λ + t + λ = t α k + β k t k = N + α + β + α 3 + β 3 t t + α k + β k t k. The specific bi-partite structure of A t enables us to write the characteristic polynoial 6 explicitly, fro which we deduce, for sufficiently large t, that both tλ + t and tλ t are zeros of the function f λ = λ + s λ t. If yt is a zero of λ + s λ t =, which is even in t, then also y t is a zero, which shows that tλ + t = tλ t. The zeros of f λ can be expanded in a Lagrange series around λ =±t. By also expanding the coefficients of this Lagrange series into a power series expansion in λ as shown in Appendix B, all coefficients α k can be coputed and we indeed find that λ + t = λ t. Hence, t λ + t If α 3 = N 3 is obtained. + λ t N N + N 3 Acknowledgeent = α k+ t k k= = N + α 3 t k=4 + α k+ t k. k= >, a tighter lower bound for λ 3 ax A than the classical λ ax A N We are very grateful to a referee for pointing us to the association with perturbation theory. Appendix A. Results fro linear algebra Fro the Schur identity [ ] [ ][ ] A B I O A B = C D CA I O D CA B we find that [ ] A B det = det A detd CA B 5 C D and D CA B is called the Schur copleent of A. 4
Lea A.. If [ ] A C X = C T B P. Van Mieghe / Linear Algebra and its Applications 47 7 9 9 5 is a real syetric atrix, where A and B are square, and consequently syetric, atrices, then λ ax X + λ in X λ ax A + λ ax B 6 Proof. See e.g. [, p. 56]. Appendix B. Characteristic coefficients of a coplex function If f z has a Taylor series around z, f z = f k z z z k with f k z = k! k= = d k f z dz k then the general relation where Gz is analytic around f z is d k Gp Gf z = Gf z + k! dp k s[k, ] f z z z z, 7 p=f z z=z where the characteristic coefficient [5] of a coplex function f z is s[k, ] f z z = ki= j i =;j i > i= which obeys the recursion relation s[,] f z z =f z, s[k, ] f z z = k+ j= k f ji z f j z s[k, j] f z z k >. 8 The zero ζz of f z closest to z is given [5] in ters of the coefficients f k z of the series expansion of f z around z as ζz = f = z f z f z n k n + k k + kf z k s [k, n ]z n= f z n, f z where s [k, ] =s[k, ] f f + denotes that the index of all Taylor coefficients appearing in 8 is augented by. Explicitly suing the first five ters n 5, 9
6 P. Van Mieghe / Linear Algebra and its Applications 47 7 9 9 ζz z f z f z f [ z f z f z + f z f z f z [ ] f z 3 + 5 + 5 f 3z f z f z f z f z f f 4z z f z f z + f 3z f z 4 ] f z 3 f z [ f z 4 + 4 + f 3z f z f3 z 3 f z f z f z f z ] 6 f 4z f z f z f z + f f 5z z 5. f z f z Appendix B.. Lagrange expansion The zero of f λ = λ + s λ t = will be coputed using the Lagrange series which can be efficiently coputed to any order with characteristic coefficients [5]. The Lagrange expansion 9 in ters of characteristic coefficients needs the Taylor coefficients of f λ around λ =±t, For n =, f n λ = n! d n dλ n f λ = λ t because λ t λ f λ = Siilarly, for n =, k= = and, k= f λ = + t and, for all n>, k= f n λ = n+ t λ n+ λ t k= N k λ k+ λ=λ. N k λ k+ = t N k λ k+ N k+ λ k. k + N k λ k+ k= = + k + N k λ k n + k Nk k λ k, 3 We now confine ourselves to coputing the zero ζλ of f λ accurate up to order Ot q, where q is fixed, but specified later. Since λ = Ot, for f n λ to be accurate up to order Ot q, we need to take in the coputation k = q n ters in the k-sus. Also, it follows that q >nderivatives or Taylor coefficients in the Lagrange series are needed. The Lagrange expansion 9 indicates that we need order expansions for f λ k and f z n. f z Both of these expansions can be given in ters of characteristic coefficients. Applying g k z = g k + n n + k g k n n s[n, ] z = n=
P. Van Mieghe / Linear Algebra and its Applications 47 7 9 9 7 to, we have, with z = λ and g = and g n = n+ N n for n>, that f λ k = k + n n + k s[n, ] gz λ n. = = n= Siilarly, applying h n z = h n + n h k n k s[k, ] z k+n to, we obtain, with z = λ and with h n = N n+ for n, that f λ n = n N n n k s[k, ] hz n + k The quotient f z f z f z = N k n f z follows by Cauchy s product rule for series, n = n N n n n n N + N λ N + n N n r n k s[k, r] hz n n k + r= N k r j j + n s[j,r] gz j j= r r q + j j + n s[j,r q] gz j q= j= q n k s[k, q] hz λ r k. Explicitly up to order Ot 3, f z n = n N n f z n n + n N n n n N k + n N j N N nn 3 + nn N N N λ j 3n + n N λ. + nn + N λ. Let us copute the zero of f λ = λ + s λ t = up to Ot 3. For q = 3/, we need for each derivative n<4, only k = 4 n ters. The corresponding Lagrange series is ζλ = λ f λ f λ f λ f λ f λ f λ [ ] f λ + + f f 3λ λ 3 + Ot 3. f λ f λ f λ
8 P. Van Mieghe / Linear Algebra and its Applications 47 7 9 9 and and We list the separate ters, f z f z = N + f λ = N f λ 4 f z f z N N λ N 3 N + 5N N + N λ, + N N λ N + Oλ, 3 = N 3 + Oλ 83 f λ f λ = N N λ + 3N = f λ = f λ 4 λ λ + N λ, λ + 3N λ λ f 3 λ f λ = N N λ + 3N λ λ = λ. Cobined yields the final result 3. Appendix C. A finite su expression for s λ Since A is syetrix, A k = Xdiagλ k j XT where the colus of the orthogonal atrix X consists of eigenvectors x j of A, A λi = diag λ X λ k j λ k X T = Xdiag X T. λ λ j Then, s λ = ua λi u T = k= x j;k j= λ λ j. 4 Unless the all-one vector u is an eigenvector of A, x j;k /=. Hence, in that case deta t λi = deta λi λ t x j;k λ λ j and no eigenvalue of A is a zero of f λ = λ t j= x j;k λ λ j. j=
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