ELA SYMMETRIC NONNEGATIVE REALIZATION OF SPECTRA ∗ RICARDO L. SOTO † , OSCAR ROJO † , JULIO MORO ‡ , AND ALBERTO BOROBIA § Abstract. A perturbation result, due to R. Rado and presented by H. Perfect in 1955, shows how to modify r eigenvalues of a matrix of order n, r ≤ n, via a perturbation of rank r, without changing any of the n - r remaining eigenvalues. This result extended a previous one, due to Brauer, on perturbations of rank r = 1. Both results have been exploited in connection with the nonnegative inverse eigenvalue problem. In this paper a symmetric version of Rado’s extension is given, which allows us to obtain a new, more general, sufficient condition for the existence of symmetric nonnegative matrices with prescribed spectrum. Key words. Symmetric nonnegative inverse eigenvalue problem. AMS subject classifications. 15A18, 15A51. 1. Introduction. The real nonnegative inverse eigenvalue problem (hereafter RNIEP) is the problem of characterizing all possible real spectra of entrywise n × n nonnegative matrices. For n ≥ 5 the problem remains unsolved. In the general case, when the possible spectrum Λ is a set of complex numbers, the problem has only been solved for n = 3 by Loewy and London [11]. The complex cases n =4 and n = 5 have been solved for matrices of trace zero by Reams [17] and Laffey and Meehan [10], respectively. Sufficient conditions or realizability criteria for the existence of a nonnegative matrix with a given real spectrum have been obtained in [25, 14, 15, 18, 8, 1, 19, 22] (see [3, §2.1] and references therein for a comprehensive survey). If we additionally require the realizing matrix to be symmetric, we have the symmetric nonnegative inverse eigenvalue problem (hereafter SNIEP). Both problems, RNIEP and SNIEP, are equivalent for n ≤ 4 (see [26]), but are different otherwise [7]. Partial results for the SNIEP have been obtained in [4, 24, 16, 21, 23] (see [3, §2.2] and references therein for more on the SNIEP). The origin of the present paper is a perturbation result, due to Brauer [2] (The- orem 2.2 below), which shows how to modify one single eigenvalue of a matrix via a rank-one perturbation, without changing any of the remaining eigenvalues. This result was first used by Perfect [14] in connection with the NIEP, and has given rise lately to a number of realizability criteria [19, 20, 22]. Closer to our approach in this paper is Rado’s 1 extension (Theorem 2.3 below) of Brauer’s result, which was used by * Received by the editors 23 June 2006. Accepted for publication 27 December 2006. Handling Editor: Michael Neumann. † Departamento deMatem´aticas, UniversidadCat´olica del Norte, Antofagasta, Casilla1280, Chile (rsoto@ucn.cl, orojo@ucn.cl). Supported by Fondecyt 1050026, Fondecyt 1040218 and Mecesup UCN0202, Chile. ‡ Departamento de Matem´aticas, Universidad Carlos III de Madrid, 28911–Legan´ es, Spain (jmoro@math.uc3m.es). Supported by the Spanish Ministeriode Ciencia y Tecnolog´ ıa through grant BFM-2003-00223. § Departamento de Matem´aticas, UNED, Madrid, Senda del Rey s/n., 28040 – Madrid, Spain (aborobia@mat.uned.es). 1 Perfect points out in [15] that both the extension and its proof are due to Professor R. Rado. 1 Electronic Journal of Linear Algebra ISSN 1081-3810 A publication of the International Linear Algebra Society Volume 16, pp. 1-18, January 2007 http://math.technion.ac.il/iic/ela