World Scientific
  • Search
Skip main navigation

Cookies Notification

We use cookies on this site to enhance your user experience. By continuing to browse the site, you consent to the use of our cookies. Learn More

System Upgrade on Tue, May 28th, 2024 at 2am (EDT)

Existing users will be able to log into the site and access content. However, E-commerce and registration of new users may not be available for up to 12 hours.
For online purchase, please visit us again. Contact us at [email protected] for any enquiries.

The singular values of the GOE by:7 (Source: Crossref)

    As a unifying framework for examining several properties that nominally involve eigenvalues, we present a particular structure of the singular values of the Gaussian orthogonal ensemble (GOE): the even-location singular values are distributed as the positive eigenvalues of a Gaussian ensemble with chiral unitary symmetry, while the odd-location singular values, conditioned on the even-location ones, can be algebraically transformed into a set of independent χ-distributed random variables. We discuss three applications of this structure: first, there is a pair of bidiagonal square matrices, whose singular values are jointly distributed as the even- and odd-location ones of the GOE; second, the magnitude of the determinant of the GOE is distributed as a product of simple independent random variables; third, on symmetric intervals, the gap probabilities of the GOE can be expressed in terms of the Laguerre unitary ensemble. We work specifically with matrices of finite order, but by passing to a large matrix limit, we also obtain new insight into asymptotic properties such as the central limit theorem of the determinant or the gap probabilities in the bulk-scaling limit. The analysis in this paper avoids much of the technical machinery (e.g. Pfaffians, skew-orthogonal polynomials, martingales, Meijer G-function, etc.) that was previously used to analyze some of the applications.

    AMSC: 15B52, 60B20, 60F05, 62E15


    • G. E. Andrews, I. P. Goulden and D. M. Jackson, Stud. Appl. Math. 110(4), 377 (2003). CrossrefGoogle Scholar
    • F. Bornemann, Markov Process. Related Fields 16(4), 803 (2010). Google Scholar
    • F. Bornemann and P. J. Forrester, Singular values and evenness symmetry in random matrix theory, preprint (2015) , arXiv:1503.07383 . Google Scholar
    • R. Delannay and G. Le Caër, Phys. Rev. E (3) 62(2), 1526 (2000). CrossrefGoogle Scholar
    • L.   Devroye , Nonuniform Random Variate Generation ( Springer-Verlag , New York , 1986 ) . CrossrefGoogle Scholar
    • I. Dumitriu and P. J. Forrester, J. Math. Phys. 51(9), 25 (2010). CrossrefGoogle Scholar
    • A. Edelman and M. La Croix, The singular values of the GUE (less is more), preprint (2014) , arXiv:1410.7065 . Google Scholar
    • P. J. Forrester, Forum Math. 18(5), 711 (2006). CrossrefGoogle Scholar
    • P. J.   Forrester , Log-Gases and Random Matrices ( Princeton University Press , Princeton, NJ , 2010 ) . CrossrefGoogle Scholar
    • P. J. Forrester and E. M. Rains, Random Matrix Models and Their Applications (Cambridge University Press, Cambridge, 2001) pp. 171–207. Google Scholar
    • P. J. Forrester and E. M. Rains, Probab. Theory Related Fields 131(1), 1 (2005). CrossrefGoogle Scholar
    • G. H. Golub and C. F. Van Loan, Matrix Computations, 4th edn. (Johns Hopkins University Press, Baltimore, 2013). CrossrefGoogle Scholar
    • R. A.   Horn and C. R.   Johnson , Matrix Analysis , 2nd edn. ( Cambridge University Press , Cambridge , 2013 ) . Google Scholar
    • I. M. Johnstone and Z. Ma, Ann. Appl. Probab. 22(5), 1962 (2012). CrossrefGoogle Scholar
    • M. L.   Mehta , Random Matrices , 3rd edn. ( Elsevier/Academic Press , Amsterdam , 2004 ) . Google Scholar
    • S.   Schechter , Math. Tables Aids Comput.   13 , 73 ( 1959 ) . CrossrefGoogle Scholar
    • T. Tao and V. Vu, Adv. Math. 231(1), 74 (2012). CrossrefGoogle Scholar
    • R. C. Thompson, Linear Algebra Appl. 13(2), 69 (1976). CrossrefGoogle Scholar