Several inequalities involving Khatri–Rao products of positive semidefinite matrices

Linear Algebra and Its Applications

Volumn 354, Issue 1-3, 2002, Pages 175-186

  Liu, Shuangzhe ^a  

^a AUSTRALIAN NATIONAL UNIVERSITY   (Australia)

Author keywords

Hadamard product; Khatri Rao product; Kronecker product; Matrix inequality; Moore Penrose inverse; Tracy Singh product

Indexed keywords

HADAMARD MATRICES;

HADAMARD; HADAMARD PRODUCTS; KRONECKER PRODUCT; MATRIX INEQUALITY; POSITIVE SEMIDEFINITE MATRICES;

MATRIX ALGEBRA;

EID: 85014085914     PISSN: 00243795     EISSN: None     Source Type: Journal    
DOI: 10.1016/S0024-3795(02)00338-5     Document Type: Article

References (31)

1. [1] Albert, A., Conditions for positive and nonnegative definiteness in terms of pseudoinverses. SIAM J. Appl. Math. 17 (1969), 434–440.
2. [2] Ando, T., Concavity of certain maps on positive definite matrices and applications to Hadamard products. Linear Algebra Appl. 26 (1979), 203–241.
3. [3] Browne, M.W., Generalized least squares estimators in the analysis of covariance structures. South African Statist. J. 8 (1974), 1–24.
4. [4] R.A. Brualdi, From the Editor-in-Chief. Comment on: “Matrix results on the Khatri–Rao and Tracy–Singh products” [Linear Algebra Appl. 289 (1999) 267–277] by S. Liu, Linear Algebra Appl. 320 (2000) 212.
5. [5] S.W. Drury, S. Liu, C.-Y. Lu, S. Puntanen, G.P.H. Styan, Some comments on several matrix inequalities with applications to canonical correlations: Historical background and recent developments, Sankhyā, the Special Issue associated with “An International Conference in Honor of Professor C.R. Rao on the Occasion of his 80th Birthday, Statistics: Reflections on the Past and Visions for the Future, March 2000, The University of Texas at San Antonio”, 2001.
6. [6] Faliva, M., Identificazione e Stima nel Modello Lineare ad Equazioni Simultanee. 1983, Vita e Pensiero, Milan, Italy.
7. [7] Horn, R.A., The Hadamard product. Proc. Symp. Appl. Math. 40 (1990), 87–169.
8. [8] Horn, R.A., Mathias, R., Block-matrix generalizations of Schur's basic theorems on Hadamard products. Linear Algebra Appl. 172 (1992), 337–346.
9. [9] Khatri, C.G., Rao, C.R., Solutions to some functional equations and their applications to characterization of probability distributions. Sankhyā 30 (1968), 167–180.
10. [10] Kollo, T., Neudecker, H., Asymptotics of Pearson–Hotelling principal-component vectors of sample variance and correlation matrices. Behaviormetrika 24:1 (1997), 51–69.
11. [11] Koning, R.H., Neudecker, H., Wansbeek, T., Block Kronecker products and the vecb operator. Linear Algebra Appl. 149 (1991), 165–184.
12. [12] Liu, S., Contributions to Matrix Calculus and Applications in Econometrics Tinbergen Institute Research Series, vol. 106, 1995, Thesis Publishers, Amsterdam, The Netherlands.
13. [13] Liu, S., Matrix results on the Khatri–Rao and Tracy–Singh products. Linear Algebra Appl. 289 (1999), 267–277.
14. [14] Liu, S., Neudecker, H., Several matrix Kantorovich-type inequalities. J. Math. Anal. Appl. 197 (1996), 23–26.
15. [15] C.-Y. Lu, Generalized matrix versions of the Wielandt inequality with some statistical applications, Department of Mathematics, Northeast Normal University, Changchun, China, 1999, 8pp (preprint).
16. [16] J.R. Magnus, H. Neudecker, Matrix Differential Calculus with Applications in Statistics and Econometrics, Wiley, Chichester, UK, 1999 (revised edition).
17. [17] Mond, B., Pečarić, J.E., Inequalities for the Hadamard product of matrices. SIAM J. Matrix Anal. Appl. 19:1 (1998), 66–70.
18. [18] H. Neudecker, personal communication, 1999.
19. [19] Pukelsheim, F., On Hsu's model in regression analysis. Statistics 8 (1977), 323–331.
20. [20] Rao, C.R., Estimation of heteroscedastic variances in linear models. J. Am. Statist. Assoc. 65 (1970), 161–172.
21. [21] Rao, C.R., Kleffe, J., Estimation of Variance Components and Applications. 1988, North-Holland, Amsterdam, The Netherlands.
22. [22] Rao, C.R., Rao, M.B., Matrix Algebra and Its Applications to Statistics and Econometrics. 1998, World Scientific, Singapore.
23. [23] Styan, G.P.H., Hadamard products and multivariate statistical analysis. Linear Algebra Appl. 6 (1973), 217–240.
24. [24] Tracy, D.S., Singh, R.P., A new matrix product and its applications in matrix differentiation. Statist. Neerlandica 26 (1972), 143–157.
25. [25] Visick, G., An algebraic relationship between the Hadamard and Kronecker product with some applications. Bull. Soc. Math. Belgique B 42:3 (1990), 275–283.
26. [26] G. Visick, A unified approach to the analysis of the Hadamard product of matrices using properties of the Kronecker product, Ph.D. Thesis, London University, UK, 1998.
27. [27] Visick, G., A quantitative version of the observation that the Hadamard product is a principal submatrix of the Kronecker product. Linear Algebra Appl. 304 (2000), 45–68.
28. [28] Wang, B.-Y., Zhang, F.-Z., Schur complements and matrix inequalities of Hadamard products. Linear and Multilinear Algebra 43 (1997), 315–326.
29. [29] Wang, S.-G., Ip, W.-C., A matrix version of the Wielandt inequality and its applications to statistics. Linear Algebra Appl. 296 (1999), 171–181.
30. [30] Zhan, X., Inequalities involving Hadamard products and unitarily invariant norms. Adv. in Math. (China) 27 (1998), 416–422.
31. [31] F.-Z. Zhang, personal communication, 1999.