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Separation theorems for singular values of matrices and their applications in multivariate analysis

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  • Rao, C. Radhakrishna

Abstract

Separation theorems for singular values of a matrix, similar to the Poincaré separation theorem for the eigenvalues of a Hermitian matrix, are proved. The results are applied to problems in approximating a given r.v. by an r.v. in a specified class. In particular, problems of canonical correlations, reduced rank regression, fitting an orthogonal random variable (r.v.) to a given r.v., and estimation of residuals in the Gauss-Markoff model are discussed. In each case, a solution is obtained by minimizing a suitable norm. In some cases a common solution is shown to minimize a wide class of norms known as unitarily invariant norms introduced by von Neumann.

Suggested Citation

  • Rao, C. Radhakrishna, 1979. "Separation theorems for singular values of matrices and their applications in multivariate analysis," Journal of Multivariate Analysis, Elsevier, vol. 9(3), pages 362-377, September.
    • Handle: RePEc:eee:jmvana:v:9:y:1979:i:3:p:362-377
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    Citations

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    Cited by:

    1. Yoshio Takane & Tadashi Shibayama, 1991. "Principal component analysis with external information on both subjects and variables," Psychometrika, Springer;The Psychometric Society, vol. 56(1), pages 97-120, March.
    2. Haruhiko Ogasawara, 2009. "Asymptotic expansions in the singular value decomposition for cross covariance and correlation under nonnormality," Annals of the Institute of Statistical Mathematics, Springer;The Institute of Statistical Mathematics, vol. 61(4), pages 995-1017, December.
    3. Cajo Braak, 1990. "Interpreting canonical correlation analysis through biplots of structure correlations and weights," Psychometrika, Springer;The Psychometric Society, vol. 55(3), pages 519-531, September.
    4. Zaka Ratsimalahelo, 2003. "Strongly Consistent Determination of the Rank of Matrix," EERI Research Paper Series EERI_RP_2003_04, Economics and Econometrics Research Institute (EERI), Brussels.
    5. J. Ramsay & Jos Berge & G. Styan, 1984. "Matrix correlation," Psychometrika, Springer;The Psychometric Society, vol. 49(3), pages 403-423, September.
    6. Dümbgen, Lutz, 1998. "Perturbation Inequalities and Confidence Sets for Functions of a Scatter Matrix," Journal of Multivariate Analysis, Elsevier, vol. 65(1), pages 19-35, April.

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