Generalized matrix versions of the Cauchy-Schwarz and Kantorovich inequalities
Rent the article at a discountRent now
* Final gross prices may vary according to local VAT.Get Access
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.
- 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.
- 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.
- Baksalary, J. K. andKala, R.,Two properties of a nonnegative definite matrix. Bull. Acad. Polon. Sci. Sér. Sci. Math.28 (1980), pp. 233–235.
- Baksalary, J. K., Kala, R., andKłaczyński, K.,The matrix inequality M ⩾ B* MB. Linear Algebra Appl.54 (1983), pp. 77–86.
- Chipman, J. S.,On least squares with insufficient observations. J. Amer. Statist. Assoc.59 (1964), pp. 1078–1111.
- Chollet, J.,On principal submatrices. Linear and Multilinear Algebra11 (1982), pp. 283–285.
- 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.
- Gaffke, N. andKrafft, O.,Optimum properties of Latin square designs and a matrix inequality. Math. Operationsforsch. Statist. Ser. Statist.8, (1977), pp. 345–350.
- 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.
- Marcus, M.,A remark on the preceding paper. Linear and Multilinear Algebra11 (1982), p. 287.
- Marsaglia, G. andStyan, G. P. H.,Rank conditions for generalized inverses of partitioned matrices. Sankhyā Ser. A36 (1974), pp. 437–442.
- 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.
- Marshall, A. W. andOlkin, I.,Inequalities: Theory of majorization and its applications. Academic Press, New York, 1979.
- Marshall, A. W. andOlkin, I.,Matrix versions of the Cauchy and Kantorovich inequalities. Aequationes Math.40 (1990), pp. 89–93.
- 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.
- Generalized matrix versions of the Cauchy-Schwarz and Kantorovich inequalities
Volume 41, Issue 1 , pp 103-110
- Cover Date
- Print ISSN
- Online ISSN
- Additional Links