×

Block Kronecker products and the vecb operator. (English) Zbl 0722.15031

This paper is concerned with two generalizations of the Kronecker product and two related generalizations of the vecb operator. The authors define a less restricted generalized Kronecker product \(A\otimes B\) of two matrices A and B. Whereas the product studied before is based on the partitioning of both matrices A and B, they presuppose in this paper a partition of B only. Researchers can partition A according to their needs and wishes.
In Section 1 the authors discuss the case of balanced partitioning. In Section 2 they elaborate the general case of unbalanced partitioning. Finally, in Section 3 they apply some of the techniques discussed.

MSC:

15A69 Multilinear algebra, tensor calculus
15A30 Algebraic systems of matrices
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] Hyland, D. C.; Collins, E. G., Block Kronecker products and block norm matrices in large-scale systems analysis, SIAM J. Matrix Anal. Appl., 10, 18-29 (1989) · Zbl 0665.15015
[2] Kapteyn, A.; Neudecker, H.; Wansbeek, T., An approach to \(n\)-mode components analysis, Psychometrika, 51, 269-275 (1986) · Zbl 0613.62078
[3] Koning, R. H.; Neudecker, H.; Wansbeek, T., A note on approximately efficient estimation of covariance structure models (1989), manuscript
[4] Magnus, J. R.; Neudecker, H., The commutation matrix: Some properties and applications, Ann. Statist., 7, 381-394 (1979) · Zbl 0414.62040
[5] Magnus, J. R.; Neudecker, H., The elimination matrix: Some lemmas and applications, SIAM J. Algebraic Discrete Methods, 1, 422-449 (1980) · Zbl 0497.15014
[6] Neudecker, H.; Wansbeek, T., Some results on commutation matrices, with statistical applications, Canad. J. Statist., 11, 221-231 (1983) · Zbl 0538.15011
[7] Singh, R. P., Some Generalizations in Matrix Differentiation with Applications in Multivariate Analysis, (Ph.D. Dissertation (1972), Univ. of Windsor)
[8] Tracy, D. S.; Jinadasa, K. G., Partitioned Kronecker products of matrices and applications, Canad. J. Statist., 17, 107-120 (1989) · Zbl 0684.62044
[9] Tracy, D. S.; Singh, R. P., A new matrix product and its applications in partitioned matrix differentiation, Statist. Neerland., 26, 143-157 (1972) · Zbl 0267.15009
[10] Verhees, J., Econometric Analysis of Multidimensional Models, (Ph.D. Dissertation (1989), Univ. of Groningen)
[11] T. Wansbeek, Singular value decomposition of design matrices in ANOVA with balanced data, Statist. Probab. Lett.; T. Wansbeek, Singular value decomposition of design matrices in ANOVA with balanced data, Statist. Probab. Lett. · Zbl 0742.62075
[12] Wansbeek, T.; Verhees, J., Models for multidimensional matrices in econometrics and psychometrics, (Coppi, R.; Bolasco, S., Multiway Data Analysis (1989), North-Holland: North-Holland Amsterdam)
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.