Foundations of Computational Mathematics

, Volume 8, Issue 6, pp 649–702 | Cite as

The Polynomial Method for Random Matrices



We define a class of “algebraic” random matrices. These are random matrices for which the Stieltjes transform of the limiting eigenvalue distribution function is algebraic, i.e., it satisfies a (bivariate) polynomial equation. The Wigner and Wishart matrices whose limiting eigenvalue distributions are given by the semicircle law and the Marčenko–Pastur law are special cases.

Algebraicity of a random matrix sequence is shown to act as a certificate of the computability of the limiting eigenvalue density function. The limiting moments of algebraic random matrix sequences, when they exist, are shown to satisfy a finite depth linear recursion so that they may often be efficiently enumerated in closed form.

In this article, we develop the mathematics of the polynomial method which allows us to describe the class of algebraic matrices by its generators and map the constructive approach we employ when proving algebraicity into a software implementation that is available for download in the form of the RMTool random matrix “calculator” package. Our characterization of the closure of algebraic probability distributions under free additive and multiplicative convolution operations allows us to simultaneously establish a framework for computational (noncommutative) “free probability” theory. We hope that the tools developed allow researchers to finally harness the power of infinite random matrix theory.


Random matrices Stochastic eigenanalysis Free probability Algebraic functions Resultants D-finite series 


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  1. 1.
    N.I. Akhiezer, The Classical Moment Problem and Some Related Questions in Analysis (Hafner, New York, 1965). Translated by N. Kemmer. MATHGoogle Scholar
  2. 2.
    A.G. Akritas, Sylvester’s forgotten form of the resultant, Fibonacci Quart. 31, 325–332 (1993). MATHMathSciNetGoogle Scholar
  3. 3.
    G. Anderson, O. Zeitouni, A law of large numbers for finite-range dependent random matrices, Preprint, September 2006.
  4. 4.
    Z.D. Bai, J.W. Silverstein, On the empirical distribution of eigenvalues of a class of large dimensional random matrices, J. Multi. Anal. 54, 175–192 (1995). MATHCrossRefMathSciNetGoogle Scholar
  5. 5.
    F. Bergeron, C. Reutenauer, Combinatorial resolution of systems of differential equations. III. A special class of differentially algebraic series, Eur. J. Comb. 11, 501–512 (1990). MATHMathSciNetGoogle Scholar
  6. 6.
    P. Biane, Free probability for probabilists, in Quantum Probability Communications, vol. XI, Grenoble, 1998 (World Scientific, River Edge, 2003), pp. 55–71. Google Scholar
  7. 7.
    P. Billingsley, Convergence of Probability Measures. Wiley Series in Probability and Statistics: Probability and Statistics, 2nd edn. (Wiley, New York, 1999). MATHGoogle Scholar
  8. 8.
    B. Collins, Product of random projections, Jacobi ensembles and universality problems arising from free probability, Probab. Theory Related Fields 133, 315–344 (2005). MATHCrossRefMathSciNetGoogle Scholar
  9. 9.
    P. Deift, T. Kriecherbauer, K.T.-R. McLaughlin, New results on the equilibrium measure for logarithmic potentials in the presence of an external field, J. Approx. Theory 95, 388–475 (1998). MATHCrossRefMathSciNetGoogle Scholar
  10. 10.
    P. Deift, T. Kriecherbauer, K.T.-R. McLaughlin, S. Venakides, X. Zhou, Uniform asymptotics for polynomials orthogonal with respect to varying exponential weights and applications to universality questions in random matrix theory, Commun. Pure Appl. Math. 52, 1335–1425 (1999). MATHCrossRefMathSciNetGoogle Scholar
  11. 11.
    P.A. Deift, Orthogonal Polynomials and Random Matrices: A Riemann–Hilbert Approach. Courant Lecture Notes in Mathematics, vol. 3 (New York University Courant Institute of Mathematical Sciences, New York, 1999). Google Scholar
  12. 12.
    W.B. Dozier, J.W. Silverstein, On the empirical distribution of eigenvalues of large dimensional information-plus-noise type matrices (2004).
  13. 13.
    I. Dumitriu, E. Rassart, Path counting and random matrix theory, Electron. J. Comb. 7, R-43 (2003). MathSciNetGoogle Scholar
  14. 14.
    P. Flajolet, R. Sedgewick, Analytic combinatorics: functional equations, rational and algebraic functions, Research Report 4103, INRIA (2001).
  15. 15.
    F. Hiai, D. Pet, The Semicircle Law, Free Random Variables and Entropy, vol. 77 (American Mathematical Society, Providence, 2000). Google Scholar
  16. 16.
    R.A. Horn, C.R. Johnson, Topics in Matrix Analysis (Cambridge University Press, Cambridge, 1991). MATHGoogle Scholar
  17. 17.
    A. Kuijlaars, K.T.-R. McLaughlin, Generic behavior of the density of states in random matrix theory and equilibrium problems in the presence of real analytic external fields, Commun. Pure Appl. Math. 53, 736–785 (2000). MATHCrossRefMathSciNetGoogle Scholar
  18. 18.
    V.A. Marčenko, L.A. Pastur, Distribution of eigenvalues in certain sets of random matrices, Mat. Sb. (N.S.) 72(114), 507–536 (1967). MathSciNetGoogle Scholar
  19. 19.
    B.D. McKay, The expected eigenvalue distribution of a large regular graph, Linear Algebra Appl. 40, 203–216 (1981). MATHCrossRefMathSciNetGoogle Scholar
  20. 20.
    R.R. Nadakuditi, Applied Stochastic Eigen-Analysis, PhD thesis, Department of Electrical Engineering and Computer Science. Massachusetts Institute of Technology, February 2007. Google Scholar
  21. 21.
    A. Nica, R. Speicher, Lectures on the Combinatorics of Free Probability. London Mathematical Society Lecture Note Series (Cambridge University Press, New York, 2006). MATHGoogle Scholar
  22. 22.
    N.R. Rao, RMTool: a random matrix and free probability calculator in MATLAB.
  23. 23.
    N.R. Rao, A. Edelman, The polynomial method for the eigenvectors of random matrices, Preprint. Google Scholar
  24. 24.
    B. Salvy, P. Zimmermann, Gfun: a Maple package for the manipulation of generating and holonomic functions in one variable, ACM Trans. Math. Softw. 20, 163–177 (1994). MATHCrossRefGoogle Scholar
  25. 25.
    J.W. Silverstein, The limiting eigenvalue distribution of a multivariate F matrix, SIAM J. Math. Anal. 16, 641–646 (1985). MATHCrossRefMathSciNetGoogle Scholar
  26. 26.
    J.W. Silverstein, Strong convergence of the empirical distribution of eigenvalues of large dimensional random matrices, J. Multivar. Anal. 55(2), 331–339 (1995). MATHCrossRefMathSciNetGoogle Scholar
  27. 27.
    J.W. Silverstein, S.-I. Choi, Analysis of the limiting spectral distribution of large-dimensional random matrices, J. Multivar. Anal. 54, 295–309 (1995). MATHCrossRefMathSciNetGoogle Scholar
  28. 28.
    R. Speicher, Free probability theory and non-crossing partitions, Sém. Lothar. Comb. 39, Art. B39c (1997) (electronic). MathSciNetGoogle Scholar
  29. 29.
    R. Speicher, Free probability theory and random matrices, In Asymptotic Combinatorics with Applications to Mathematical Physics, St. Petersburg, 2001. Lecture Notes in Mathematics, vol. 1815, pp. 53–73 (Springer, Berlin, 2003). CrossRefGoogle Scholar
  30. 30.
    R.P. Stanley, Enumerative Combinatorics, vol. 2. Cambridge Studies in Advanced Mathematics, vol. 62 (Cambridge University Press, Cambridge, 1999). With a foreword by Gian-Carlo Rota and Appendix 1 by Sergey Fomin. Google Scholar
  31. 31.
    B. Sturmfels, Introduction to resultants, in Applications of Computational Algebraic Geometry, San Diego, CA, 1997. Proceedings of Symposia in Applied Mathematics, vol. 53 (American Mathematical Society, Providence, 1998), pp. 25–39. Google Scholar
  32. 32.
    A.M. Tulino, S. Verdú, Random matrices and wireless communications, Found. Trends Commun. Inf. Theory 1 (2004). Google Scholar
  33. 33.
    D. Voiculescu, Symmetries of some reduced free product C *-algebras, in Operator Algebras and Their Connections with Topology and Ergodic Theory, Buşteni, 1983. Lecture Notes in Mathematics, vol. 1132 (Springer, Berlin, 1985), pp. 556–588. CrossRefGoogle Scholar
  34. 34.
    D. Voiculescu, Addition of certain noncommuting random variables, J. Funct. Anal. 66, 323–346 (1986). MATHCrossRefMathSciNetGoogle Scholar
  35. 35.
    D. Voiculescu, Multiplication of certain noncommuting random variables, J. Oper. Theory 18, 223–235 (1987). MATHMathSciNetGoogle Scholar
  36. 36.
    D. Voiculescu, Limit laws for random matrices and free products, Invent. Math. 104, 201–220 (1991). MATHCrossRefMathSciNetGoogle Scholar
  37. 37.
    D.V. Voiculescu, K.J. Dykema, A. Nica, Free Random Variables. CRM Monograph Series, vol. 1 (American Mathematical Society, Providence, 1992). MATHGoogle Scholar
  38. 38.
    E.P. Wigner, Characteristic vectors of bordered matrices with infinite dimensions, Ann. Math. 62, 548–564 (1955). CrossRefMathSciNetGoogle Scholar

Copyright information

© SFoCM 2007

Authors and Affiliations

  1. 1.MIT Department of Electrical Engineering and Computer ScienceCambridgeUSA
  2. 2.MIT Department of MathematicsCambridgeUSA

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