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Some remarks on operators preserving partial orders of matrices

100%
Hauke J.
Discussiones Mathematicae Probability and Statistics
|
2008
|
tom 28
|
nr 1
143-155
EN
Stępniak [Linear Algebra Appl. 151 (1991)] considered the problem of equivalence of the Löwner partial order of nonnegative definite matrices and the Löwner partial order of squares of those matrices. The paper was an important starting point for investigations of the problem of how an order between two matrices A and B from different sets of matrices can be preserved for the squares of the corresponding matrices A² and B², in the sense of the Löwner partial ordering, the star partial ordering, the minus partial ordering, and the sharp partial ordering. Many papers have since been published (mostly coauthored by J.K. Baksalary - to whom the present paper is dedicated) that generalize the results in two directions: by widening the class of matrices considered and by replacing the squares by arbitrary powers. In the present paper we make a résumé of some of these results and suggest some further generalizations for polynomials of the matrices considered.
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The generalization of some properties of universal matrices

63%
Hauke J., Krzyszkowska J.
Mathematica Applicanda
|
1991
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tom 20
|
nr 34
PL
.
EN
We consider an k x k universal matrix G introduced by Hellwig (1976). We analyze two problems: the nonsingularity of the matrix G and the sign-consistency of the elements of the vector r = (ri) and of the solutrion b = (bi) to the system Gb = r. Obtained results generalize the results presented by Kolupa (1977, 1980).
3
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On orderings induced by the Loewner partial ordering

63%
Hauke J., Markiewicz A.
Applicationes Mathematicae
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1993-1995
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tom 22
|
nr 2
145-154
EN
The partial ordering induced by the Loewner partial ordering on the convex cone comprising all matrices which multiplied by a given positive definite matrix become nonnegative definite is considered. Its relation to orderings which are induced by the Loewner partial ordering of the squares of matrices is presented. Some extensions of the latter orderings and their comparison to star orderings are given.
4
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Applications of saddle-point determinants

51%
Hauke J., Johnson C., Ostrowski T.
Discussiones Mathematicae - General Algebra and Applications
|
2015
|
tom 35
|
nr 2
213-220
EN
For a given square matrix $A ∈ M_n({ℝ})$ and the vector $e ∈ (ℝ)^{n}$ of ones denote by (A,e) the matrix ⎡ A e ⎤ ⎣ $e^{T}$ 0 ⎦ This is often called the saddle point matrix and it plays a significant role in several branches of mathematics. Here we show some applications of it in: game theory and analysis. An application of specific saddle point matrices that are hollow, symmetric, and nonnegative is likewise shown in geometry as a generalization of Heron's formula to give the volume of a general simplex, as well as a conditions for its existence.
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