aequationes mathematicae

, Volume 41, Issue 1, pp 103–110 | Cite as

Generalized matrix versions of the Cauchy-Schwarz and Kantorovich inequalities

  • J. K. Baksalary
  • S. Puntanen
Research Papers


A recent note by Marshall and Olkin (1990), in which the Cauchy-Schwarz and Kantorovich inequalities are considered in matrix versions expressed in terms of the Loewner partial ordering, is extended to cover positive semidefinite matrices in addition to positive definite ones.

AMS (1980) subject classification

Primary 15A42 Secondary 15A45 


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  1. [1]
    Baksalary, J. K.,Algebraic characterizations and statistical implications of the commutativity of orthogonal projectors. InProceedings of the Second International Tampere Conference in Statistics (Pukkila, T. and Puntanen, S., Eds.). Department of Mathematical Sciences, University of Tampere, Tampere, 1987, pp. 113–142.Google Scholar
  2. [2]
    Baksalary, J. K. andKala, R.,Relationships between some representations of the best linear unbiased estimator in the general Gauss-Markoff model. SIAM J. Appl. Math.35 (1978), pp. 515–520.Google Scholar
  3. [3]
    Baksalary, J. K. andKala, R.,Two properties of a nonnegative definite matrix. Bull. Acad. Polon. Sci. Sér. Sci. Math.28 (1980), pp. 233–235.Google Scholar
  4. [4]
    Baksalary, J. K., Kala, R., andKłaczyński, K.,The matrix inequality M ⩾ B* MB. Linear Algebra Appl.54 (1983), pp. 77–86.Google Scholar
  5. [5]
    Chipman, J. S.,On least squares with insufficient observations. J. Amer. Statist. Assoc.59 (1964), pp. 1078–1111.Google Scholar
  6. [6]
    Chollet, J.,On principal submatrices. Linear and Multilinear Algebra11 (1982), pp. 283–285.Google Scholar
  7. [7]
    Cline, R. E. andGreville, T. N. E.,An extension of the generalized inverse of a matrix. SIAM J. Appl. Math.19 (1970), pp. 682–688.Google Scholar
  8. [8]
    Gaffke, N. andKrafft, O.,Optimum properties of Latin square designs and a matrix inequality. Math. Operationsforsch. Statist. Ser. Statist.8, (1977), pp. 345–350.Google Scholar
  9. [9]
    Magness, T. A. andMcGuire, J. B.,Comparison of least squares and minimum variance estimates of regression parameters. Ann. Math. Statist.33 (1962), pp. 462–470.Google Scholar
  10. [10]
    Marcus, M.,A remark on the preceding paper. Linear and Multilinear Algebra11 (1982), p. 287.Google Scholar
  11. [11]
    Marsaglia, G. andStyan, G. P. H.,Rank conditions for generalized inverses of partitioned matrices. Sankhyā Ser. A36 (1974), pp. 437–442.Google Scholar
  12. [12]
    Marshall, A. W. andOlkin, I.,Reversal of the Lyapunov, Hölder, and Minkowski inequalities and other extensions of the Kantorovich inequality. J. Math. Anal. Appl.8 (1964), pp. 503–514.Google Scholar
  13. [13]
    Marshall, A. W. andOlkin, I.,Inequalities: Theory of majorization and its applications. Academic Press, New York, 1979.Google Scholar
  14. [14]
    Marshall, A. W. andOlkin, I.,Matrix versions of the Cauchy and Kantorovich inequalities. Aequationes Math.40 (1990), pp. 89–93.Google Scholar
  15. [15]
    Rao, C. R.,Least squares theory using an estimated dispersion matrix and its application to measurement of signals. InProceedings of the Fifth Berkeley Symposium on Mathematical Statistics and Probability, Vol. 1 (Le Cam, L. M. and Neyman, J., Eds.). University of California Press, Berkeley, CA, 1967, pp. 355–372.Google Scholar

Copyright information

© Birkhäuser Verlag 1991

Authors and Affiliations

  • J. K. Baksalary
    • 1
    • 2
  • S. Puntanen
    • 1
    • 2
  1. 1.Department of MathematicsTadeusz Kotarbiński Pedagogical UniversityZielona GóraPoland
  2. 2.Department of Mathematical SciencesUniversity of TampereTampereFinland

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