×

zbMATH — the first resource for mathematics

The eigenvalues and eigenvectors of finite, low rank perturbations of large random matrices. (English) Zbl 1226.15023
Let \(X_n\) be an \(n\times n\) Hermitian or symmetric random matrix. Let \(P_n\) be an \(n\times n\) Hermitian or symmetric matrix of rank \(r\). The authors study the behaviour of the eigenvalues and eigenvectors of perturbations of \(X_n\) by \(P_n\), namely \(X_n+P_n\), \(X_n(I_n+P_n)\), \((I_n+P_n)^{1/2}X_n(I_n+P_n)^{1/2}\). Almost sure convergence of the extreme eigenvalues and of the projections of the corresponding eigenvectors on the eigenspaces of \(P_n\) are proven.
The limiting eigenvalue is shown to depend explicitly on the limiting eigenvalue distribution of \(X_n\). A threshold is found where the limit as \(n\to\infty\) of the extreme eigenvalues of the perturbed matrix differ from those of \(X_n\) if and only if the eigenvalues of \(P_n\) are above that threshold. An analogous phase transition is found for the eigenvectors.

MSC:
15B52 Random matrices (algebraic aspects)
15A18 Eigenvalues, singular values, and eigenvectors
46L54 Free probability and free operator algebras
60B20 Random matrices (probabilistic aspects)
62H25 Factor analysis and principal components; correspondence analysis
82B26 Phase transitions (general) in equilibrium statistical mechanics
Software:
RMTool
PDF BibTeX XML Cite
Full Text: DOI
References:
[1] Anderson, G.; Guionnet, A.; Zeitouni, O., An introduction to random matrices, Cambridge stud. adv. math., vol. 118, (2010), Cambridge University Press · Zbl 1184.15023
[2] Arbenz, P.; Gander, W.; Golub, G.H., Restricted rank modification of the symmetric eigenvalue problem: theoretical considerations, Linear algebra appl., 104, 75-95, (1988) · Zbl 0652.15004
[3] Bai, Z.D.; Silverstein, J., Spectral analysis of large dimensional random matrices, (2009), Springer New York
[4] Baik, J.; Ben Arous, G.; Péché, S., Phase transition of the largest eigenvalue for nonnull complex sample covariance matrices, Ann. probab., 33, 5, 1643-1697, (2005) · Zbl 1086.15022
[5] Baik, J.; Silverstein, J.W., Eigenvalues of large sample covariance matrices of spiked population models, J. multivariate anal., 97, 6, 1382-1408, (2006) · Zbl 1220.15011
[6] Bassler, K.E.; Forrester, P.J.; Frankel, N.E., Eigenvalue separation in some random matrix models, J. math. phys., 50, 3, 033302, (2009), 24 pp · Zbl 1202.82045
[7] Benaych-Georges, F., Infinitely divisible distributions for rectangular free convolution: classification and matricial interpretation, Probab. theory related fields, 139, 1-2, 143-189, (2007) · Zbl 1129.15019
[8] Benaych-Georges, F., Rectangular random matrices, related convolution, Probab. theory related fields, 144, 3, 471-515, (2009) · Zbl 1171.15022
[9] Benaych-Georges, F., Rectangular R-transform at the limit of rectangular spherical integrals, (2009), available online at
[10] Benaych-Georges, F., On a surprising relation between the marchenko-Pastur law, rectangular and square free convolutions, Ann. inst. H. Poincaré probab. statist., 46, 3, 644-652, (2010) · Zbl 1206.46055
[11] Benaych-Georges, F.; Guionnet, A.; Maïda, M., Fluctuations of the extreme eigenvalues of finite rank deformations of random matrices, (2010), available online at · Zbl 1261.15042
[12] Benaych-Georges, F.; Guionnet, A.; Maïda, M., Large deviations of the extreme eigenvalues of finite rank deformations of random matrices, (2010), available online at · Zbl 1261.15042
[13] F. Benaych-Georges, R.R. Nadakuditi, The extreme singular values and singular vectors of finite, low rank perturbations of large random rectangular matrices, in preparation. · Zbl 1252.15039
[14] Bunch, J.R.; Nielsen, C.P.; Sorensen, D.C., Rank-one modification of the symmetric eigenproblem, Numer. math., 31, 1, 31-48, (1978/1979) · Zbl 0369.65007
[15] Capitaine, M.; Donati-Martin, C.; Féral, D., The largest eigenvalues of finite rank deformation of large Wigner matrices: convergence and nonuniversality of the fluctuations, Ann. probab., 37, 1, 1-47, (2009) · Zbl 1163.15026
[16] Collins, B., Product of random projections, Jacobi ensembles and universality problems arising from free probability, Probab. theory related fields, 133, 3, 315-344, (2005) · Zbl 1100.46036
[17] Collins, B.; Śniady, P., Integration with respect to the Haar measure on unitary, orthogonal and symplectic group, Comm. math. phys., 264, 3, 773-795, (2006) · Zbl 1108.60004
[18] Deift, P.; Kriecherbauer, T.; McLaughlin, K.T.-R.; Venakides, S.; Zhou, X., Uniform asymptotics for polynomials orthogonal with respect to varying exponential weights and applications to universality questions in random matrix theory, Comm. pure appl. math., 52, 1335-1425, (1999) · Zbl 0944.42013
[19] El Karoui, N., Tracy-Widom limit for the largest eigenvalue of a large class of complex sample covariance matrices, Ann. probab., 35, 2, 663-714, (2007) · Zbl 1117.60020
[20] Féral, D.; Péché, S., The largest eigenvalue of rank one deformation of large Wigner matrices, Comm. math. phys., 272, 1, 185-228, (2007) · Zbl 1136.82016
[21] Guionnet, A.; Maïda, M., Character expansion method for the first order asymptotics of a matrix integral, Probab. theory related fields, 132, 4, 539-578, (2005) · Zbl 1113.60031
[22] Guionnet, A.; Maïda, M., A Fourier view on the R-transform and related asymptotics of spherical integrals, J. funct. anal., 222, 2, 435-490, (2005) · Zbl 1065.60023
[23] Hiai, F.; Petz, D., The semicircle law, free random variables and entropy, Math. surveys monogr., vol. 77, (2000), Amer. Math. Soc. Providence, RI · Zbl 0955.46037
[24] Horn, R.A.; Johnson, C.R., Matrix analysis, (1985), Cambridge University Press Cambridge · Zbl 0576.15001
[25] Hoyle, D.C.; Rattray, M., Statistical mechanics of learning multiple orthogonal signals: asymptotic theory and fluctuation effects, Phys. rev. E (3), 75, 1, 016101, (2007), 13 pp
[26] Ipsen, I.C.F.; Nadler, B., Refined perturbation bounds for eigenvalues of Hermitian and non-Hermitian matrices, SIAM J. matrix anal. appl., 31, 1, 40-53, (2009) · Zbl 1189.15022
[27] Kuijlaars, A.; McLaughlin, K.T.-R., Generic behavior of the density of states in random matrix theory and equilibrium problems in the presence of real analytic external fields, Comm. pure appl. math., 53, 736-785, (2000) · Zbl 1022.31001
[28] Ledoux, M., The concentration of measure phenomenon, (2001), Amer. Math. Soc. Providence, RI · Zbl 0995.60002
[29] Marčenko, V.A.; Pastur, L.A., Distribution of eigenvalues in certain sets of random matrices, Mat. sb. (N.S.), 72, 114, 507-536, (1967) · Zbl 0152.16101
[30] Nadakuditi, R.R.; Silverstein, J.W., Fundamental limit of sample generalized eigenvalue based detection of signals in noise using relatively few signal-bearing and noise-only samples, J. selected top. sig. processing, 4, 3, 468-480, (2010)
[31] Nadler, B., Finite sample approximation results for principal component analysis: a matrix perturbation approach, Ann. statist., 36, 6, 2791-2817, (2008) · Zbl 1168.62058
[32] Paul, D., Asymptotics of sample eigenstructure for a large dimensional spiked covariance model, Statist. sinica, 17, 4, 1617-1642, (2007) · Zbl 1134.62029
[33] Péché, S., The largest eigenvalue of small rank perturbations of Hermitian random matrices, Probab. theory related fields, 134, 1, 127-173, (2006) · Zbl 1088.15025
[34] Rao, N.R.; Edelman, A., The polynomial method for random matrices, Found. comput. math., 8, 6, 649-702, (2008) · Zbl 1171.15024
[35] Silverstein, J.W., Some limit theorems on the eigenvectors of large-dimensional sample covariance matrices, J. multivariate anal., 15, 3, 295-324, (1984) · Zbl 0553.60011
[36] Silverstein, J.W., On the eigenvectors of large-dimensional sample covariance matrices, J. multivariate anal., 30, 1, 1-16, (1989) · Zbl 0678.60011
[37] Silverstein, J.W., Weak convergence of random functions defined by the eigenvectors of sample covariance matrices, J. multivariate anal., 18, 3, 1-16, (1990)
[38] Silverstein, J.W.; Choi, S.-I., Analysis of the limiting spectral distribution of large-dimensional random matrices, J. multivariate anal., 54, 2, 295-309, (1995) · Zbl 0872.60013
[39] Stewart, G.W.; Sun, J.G., Matrix perturbation theory, Comput. sci. sci. comput., (1990), Academic Press Inc. Boston, MA
[40] Voiculescu, D.V.; Dykema, K.J.; Nica, A., Free random variables, CRM monogr. ser., vol. 1, (1992), Amer. Math. Soc. Providence, RI · Zbl 0795.46049
[41] Wigner, E.P., On the distribution of the roots of certain symmetric matrices, Ann. of math. (2), 67, 325-327, (1958) · Zbl 0085.13203
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.