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Artykuł dostępny w postaci pełnego tekstu - kliknij by otworzyć plik
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The vector cross product from an algebraic point of view

100%
Trenkler G.
Discussiones Mathematicae - General Algebra and Applications
|
2001
|
tom 21
|
nr 1
67-82
EN
The usual vector cross product of the three-dimensional Euclidian space is considered from an algebraic point of view. It is shown that many proofs, known from analytical geometry, can be distinctly simplified by using the matrix oriented approach. Moreover, by using the concept of generalized matrix inverse, we are able to facilitate the analysis of equations involving vector cross products.
2
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On melancholic magic squares

63%
Trenkler G., Trenkler D.
Discussiones Mathematicae Probability and Statistics
|
2013
|
tom 33
|
nr 1-2
111-119
EN
Starting with Dürer's magic square which appears in the well-known copper plate engraving Melencolia we consider the class of melancholic magic squares. Each member of this class exhibits the same 86 patterns of Dürer's magic square and is magic again. Special attention is paid to the eigenstructure of melancholic magic squares, their group inverse and their Moore-Penrose inverse. It is seen how the patterns of the original Dürer square to a large extent are passed down also to the inverses of the melancholic magic squares.
3
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Estimation of the Kronecker and products of two mean vectors in multivariate analysis

63%
Neudecker H., Trenkler G.
Discussiones Mathematicae Probability and Statistics
|
2005
|
tom 25
|
nr 2
207-215
EN
In this paper the Kronecker and inner products of mean vectorsof two different populations are considered. Using the generalized jackknife approach, estimators for these products are constructed which turn out to be unbiased, provided one can assume multinormal distribution.
4
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An alternative approach to characterize the commutativity of orthogonal projectors

63%
Baksalary O., Trenkler G.
Discussiones Mathematicae Probability and Statistics
|
2008
|
tom 28
|
nr 1
113-137
EN
In an invited paper, Baksalary [Algebraic characterizations and statistical implications of the commutativity of orthogonal projectors. In: T. Pukkila, S. Puntanen (Eds.), Proceedings of the Second International Tampere Conference in Statistics, University of Tampere, Tampere, Finland, [2], pp. 113-142] presented 45 necessary and sufficient conditions for the commutativity of a pair of orthogonal projectors. Basing on these results, he discussed therein also statistical aspects of the commutativity with reference to problems concerned with canonical correlations and with comparisons between estimators and between sets of linearly sufficient statistics corresponding to different linear models. In the present paper, parts of this analysis are resumed in order to shed some additional light on the problem of commutativity. The approach utilized is different than the one used by Baksalary, and is based on representations of projectors in terms of partitioned matrices. The usefulness of such representations is demonstrated by reinvestigating some of Baksalary's statistical considerations.
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