Convexity and Log Convexity for the Spectral Radius

The starting point of this paper is a theorem by J. F. C. Kingman which asserts that if the entries of a nonnegative matrix are log convex functions of a variable then so is the spectral radius of the matrix. A related result of J. Cohen asserts that the spectral radius of a nonnegative matrix is a convex function of the diagonal elements. The first section of this paper gives a new, unified proof of these results and also analyzes exactly when one has strict convexity. The second section gives some very simple proofs of results of Friedland and Karlin concerning “min-max” characterizations of the spectral radius of nonnegative matrices. These arguments also yield, as will be shown in another paper, min-max characterizations of the principal eigenvalue of second order elliptic boundary value problems on bounded domains. The third section considers the cone K of nonnegative vectors in R” and continuous maps f: K -+ K which are homogeneous of degree one and preserve the partial order induced by K. The (cone) spectral radius of such maps is defined and a direct generalization of Kingman’s theorem to a subclass of such nonlinear maps is given. The final section of this paper treats a problem that arises in population biology. If K, denotes the interior of K and f is as above, when can one say that f has a unique eigenvector (to within normalization) in K,? A subtle point to be noted is that f may have other eigenvectors in the boundary of K. If u E K, is an eigenvector of f, Iuj = 1, and g(x) = f( r)/lf(x)l, when can one say that for any x E K,, gP(x), the pth iterate of g acting on x, converges geometrically to u? The fourth section provides answers to these questions that are adequate for many of the population biology problems. *Partially supported by NSF MCS 82-101316 and as a visiting member of the Courant Institute, 1983-84. LINEAR ALGEBRA AND ITS APPLZCATZONS 73:59122 (1986) 59 0 Elsevier Science Publishing Co., Inc., 1986 52 Vanderbilt Ave., New York, NY 10017 0024.3795/86/$3.50 60 ROGER D. NUSSBAUM INTRODUCTION The spectral radius r(A) of a square matrix A is the maximum of { IX] : h an eigenvalue of A}. If A = (aij) is a “nonnegative matrix” (so aij >, 0 for 1 < i, j < n), Cohen [7, 81 has shown that r(A) is a convex function of the diagonal elements of A, and Kingman [17] has proved that if the entries of A are log convex functions of a parameter t, then r(A) is also a log convex function of t. Friedland [12] has also given related results concerning the convex dependence of r(A) on various parameters. In the first section of this paper we shall present a simple and unified approach to refinements of the Cohen, Kingman and Friedland theorems. In particular we shall obtain necessary and sufficient conditions for strict convexity or strict log convexity to hold in our theorems. With the partial exception of some results in [12], such necessary and sufficient conditions are inaccessible by previous methods. The second section of this paper presents a very simple approach to a “minimax” variational formula (obtained by Friedland) for the spectral radius of a nonnegative matrix A. We also give a simple proof of an earlier, closely related theorem of Friedland and Karlin [13]. We should remark that at least part of Friedland’s theorem is a consequence of an earlier, more general result of Donsker and Varadhan [ll]. However, Friedland’s result is sharper, for it explicitly gives the saddlepoint at which the minimax is achieved. Our real interest in the results of Section 2 is that the proofs can be generalized to the context of second order elliptic eigenvalue problems like hb) = Caij%,x, + xbiux, + cu = Au on 8, au+p*vu=o on ask (0.1) Here A is assumed uniformly elliptic on a smooth, bounded domain Q; a i j( x), b,(r), and c(x) are Holdercontinuous; o(r) > 0 on a!$ and p(x) is an outward-pointing vector on a&? or p(x) = 0 [in which case o(r) is assumed positive on as2]. We shall prove in [26] that variants of arguments like those in Section 2 can be used to give a variational characterization of the principal eigenvalue of (0.1). The third section of this paper is concerned with nonlinear generalizations of the Cohen and Kingman theorems. Let K = { x E R n : xi > 0 for all i } and K,= {xElw”: xi > 0 for all i } (this notation is maintained throughout this paper), and suppose f: K -+ K is a continuous map which is homogeneous of degree one [ f( Xx) = A f(x) for x E K and X > 0] and order-preserving (with respect to the partial ordering induced by x < y if y x E K). One can define eigenvectors and eigenvalues in the usual way for such maps and CONVEXITY AND LOG CONVEXITY 61 define r(f) = sup{ A >, 0 : X is an eigenvalue of f }. We call r(f) the spectral radius of 5 Our operators will depend in a natural way on certain parameters; if these parameters are, in turn, log convex functions of a variable t, we shall prove (directly generalizing Kingman’s theorem) that the generalized spectral radius is a log convex function of t. We shall also give a direct generalization in this framework of Cohen’s theorem. One point should be emphasized here: The approach which we give to the linear questions of Section 1 generalizes directly to the nonlinear context of Section 3, but other approaches to the linear theory do not seem to generalize to this framework. Eigenvectors of f in K, are equivalent to fixed points of g(x) = f( x)/lf(x)l in K, (where IuI denotes a suitable norm). It is natural to ask whether g has a unique fixed point u in K, and whether, for any x E K,, g”(x), the pth iterate of g acting on x, converges to u. In Section 4 we consider such questions and obtain theorems which reduce in the linear case to the Perron-Frobenius theorem and to a theorem of Birkhoff [3]. Corollaries 4.6 and 4.7 below are very special cases of our results which nevertheless convey the flavor of our theorems. One point should be strongly emphasized about the results in Section 4: If there exists an integer p such that fP( K (0)) c K,, then the arguments given in Section 4 can be simplified enormously. However, precisely this condition fails in many examples of interest to us, e.g., in Corollaries 4.6 and 4.7. We should also remark that the results of Section 4 are of interest in studying so-called “twesex models” in population biology, and the particular class of functions f we emphasize is motivated by examples in the population-biology literature. 1. CONVEXITY AND LOG CONVEXITY FOR THE SPECTRAL RADIUS Our prerequisites for this section comprise only some elementary facts from the theory of nonnegative matrices (see [29, Chapter 11). If A = (a ij) and B = ( bi j) are n x n matrices, we shall write A > B if a, j > bi j for 1 < i, j < n, and A > B if a ii > bi j for i < i, j < n. Analogously, if x, y E Iw “, we shall write x: >, y if xi >, yi for 1~ i < n and x > y if xi > yi for 1~ i < n; we set K={xER”: x >, 0). An n x n, nonnegative matrix A is called “irreducible” if for each pair of integers (i, j) with 1 < i, j < n, there exists an integer m = m(i, j) such that the entry in row i and column j of A”’ is positive. 62 ROGER D. NUSSBAUM The Perron-Frobenius theorem [29, pp. 20, 251 asserts that if A >, 0 (A a square matrix) and r = r(A), there exists u E K (0) such that Au = ru. Furthermore, one can easily prove directly or obtain from [29] LEMMA 1.1 (See Theorem 1.6 in [29]). Suppose that A is a nonnegative, irreducible square matrix and that there exists u E K (0) and a real number p such that Au < pu. (1.1) Then one has p > 0, u > 0, and r(A) < p. Furthermore, r(A) < p unless equality holds in (1.1). As an immediate consequence of Lemma 1.1 we have the following simple but useful observation. LEMMA 1.2. Zf A and B are n x n irreducible, nonnegative matrices such that A < B and A f B, then r(A) -C r(B). Proof. Lemma 1.1 and the Perron-Frobenius theorem imply that there exists v > 0 such that Bv = r(B)v. Because A < B, it follows that and because A f B and v > 0, equality cannot hold in the previous inequality. Lemma 1.1 now implies that r(A) < r(B). If A = (a i j) is a matrix with nonnegative off-diagonal elements (so a i j 2 0 for all i f j), we shall say, following notation in [lo], that A is essentially nonnegative. Seneta calls such matrices k&matrices; see [29, p. 401. If A is essentially nonnegative, the Perron-Frobenius theorem implies that A has a real eigenvalue h 1 = h 1( A) with corresponding eigenvector in K and RehgX,(A) CONVEXITY AND LOG CONVEXITY 63 for every other eigenvalue h of A; X,(A) is the principal eigenvalue of A. If I denotes the identity matrix and A + aZ > 0, the Perron-Frobenius theorem implies X,(A) = r(A + aZ) a. Throughout this paper, r(A) and X,(A) will denote the spectral radius and principal eigenvalue respectively of a matrix A. If A = (aij) is an essentially nonnegative matrix, define a nonnegative matrix B = (bij) by bij = aij for i # j and bii = 0, and say that A is irreducible if B is irreducible. We leave it to the reader to check that this definition agrees with the previous one when A is nonnegative. Equivalently, A is irreducible if A + aZ is irreducible whenever A + al 2 0. We need also to recall some definitions. If B is a matrix such that bij = 0 for i # j, then Z? will be called a diagonal matrix and we shall write B = diag(bii); B is a positive diagonal matrix if B is diagonal and bii > 0 for l is an n X n diagonal matrix. For 1~ i, j < n assume that xi(t) is either identically zero or a log convex jkction of t and that g,,(t) is a convex function for l, and 64 ROGER D. NUSSBAUM G(l)=diag(d,,), anddefineamutrixM(t)=(mij(t)) by (l-t)cii+tdii+af;‘btl for i=j for i# j.