IEEE TRANSACTIONS ON INFORMATION THEORY, VOL. 54, NO. 11, NOVEMBER 2008 5113 Improved Estimation of Eigenvalues and Eigenvectors of Covariance Matrices Using Their Sample Estimates Xavier Mestre, Member, IEEE Abstract—The problem of estimating the eigenvalues and eigen- vectors of the covariance matrix associated with a multivariate sto- chastic process is considered. The focus is on finite sample size situ- ations, whereby the number of observations is limited and compa- rable in magnitude to the observation dimension. Using tools from random matrix theory, and assuming a certain eigenvalue split- ting condition, new estimators of the eigenvalues and eigenvectors of the covariance matrix are derived, that are shown to be con- sistent in a more general asymptotic setting than the traditional one. Indeed, these estimators are proven to be consistent, not only when the sample size increases without bound for a fixed observa- tion dimension, but also when the observation dimension increases to infinity at the same rate as the sample size. Numerical evalua- tions indicate that the estimators have an excellent performance in small sample size scenarios, where the observation dimension and the sample size are comparable in magnitude. Index Terms—Eigenvalues, eigenvectors, G-estimation, random matrix theory, sample covariance matrix. I. INTRODUCTION E IGENVALUES and eigenvectors of covariance matrices are extensively used in multiple applications of signal pro- cessing, pattern classification, econometrics, decision theory, and statistical inference, among other fields. There exist mul- tiple estimation and detection procedures, such as direction of arrival detection [1], channel estimation [2], multiuser detection [3], or model order selection [4], which are precluded by the es- timation of the eigenvalues and eigenvectors of the covariance matrix of the observation. In all these procedures, it is important to depart from a good estimation of the eigenvalues and/or the eigenvectors of the observation covariance matrix, which is in practice unknown. The problem of estimating the eigenvalues and eigenvectors of a covariance matrix has usually been solved using the sample covariance matrix constructed from the observations. The eigen- values and eigenvectors of the true covariance matrix are usu- ally estimated as the eigenvalues and eigenvectors of the sample covariance matrix, which in what follows will be referred to as Manuscript received July 20, 2006; revised July 15, 2008. Current version published October 22, 2008. This work was supported in part by the Catalan Goverment under Grant SGR2005-00690 and the European Commission under Project NEWCOM++ 7FP-ICT-216715. The material in this paper was pre- sented in part at the 2006 Second International Symposium on Communications and Control, Marrekech, Morocco, March 2006. The author is with the Centre Tecnològic de Telecomunicacions de Catalunya Parc Mediterrani de la Tecnologia, Castelldefels (Barcelona), Spain (e-mail: xavier.mestre@cttc.cat). Communicated by X. Wang, Associate Editor for Detection and Estimation. Color versions of Figures 1 and 4–7 in this paper are available online at http:// ieeexplore.ieee.org. Digital Object Identifier 10.1109/TIT.2008.929938 sample eigenvalues and sample eigenvectors, respectively. Al- though these estimators are consistent for large sample sizes, it turns out that they have serious deficiencies in the small sample size regime, where the sample size is comparable in magnitude to the observation dimension. It is well known, for instance, that the sample eigenvalues tend to be more spread out than the original ones, so that the largest (resp., the smallest) sample eigenvalue tends to overestimate (resp., underestimate) the cor- responding eigenvalue of the true covariance matrix. Let us briefly introduce the problem formulation in its full generality. To that effect, we consider a collection of indepen- dent and identically distributed (i.i.d.) observations of a certain -dimensional stochastic process, denoted by , where , . We assume that these obser- vations have zero mean and covariance matrix . We will denote by the set of pairwise dif- ferent eigenvalues of the covariance matrix , where here is the number of distinct true eigenvalues . Each of the eigenvalues has multiplicity , , so that . Associated with each eigenvalue , , there is a complex subspace of dimension . This subspace is determined by an matrix of eigen- vectors, denoted by , such that . Note that this specification is unique up to right multiplication by an or- thogonal matrix. Hence, we can write . . . where contains all the eigenvector matrices, namely, . Now, since the eigenvector matrices associated to a particular eigenvalue are defined up to right multiplication by an orthogonal matrix, it is more convenient to formulate the problem in terms of eigenvector orthogonal projection matrices, defined as Contrary to the eigenvector matrices, the entries of these pro- jection matrices are always univocally specified. Furthermore, one can easily recover from using standard algebraic methods. Problem statement: Assuming that the multiplicities of the eigenvalues are known, estimate the true eigenvalues and their associated eigenvector projection matrices using the observations . 0018-9448/$25.00 © 2008 IEEE