scholarly journals Preservers of partial orders on the set of all variance-covariance matrices

Filomat ◽  
2021 ◽  
Vol 35 (2) ◽  
pp. 617-632
Author(s):  
Iva Golubic ◽  
Janko Marovt
Keyword(s):  
Partial Order ◽  
Positive Semidefinite ◽  
Covariance Matrices ◽  
Partial Orders ◽  
Equivalent Definition ◽  
Definition Of ◽  
Square Complex ◽  
Additive Maps ◽  
Minus Partial Order ◽  
Real Matrices

Let H+n(R) be the cone of all positive semidefinite n x n real matrices. Two of the best known partial orders that were mostly studied on subsets of square complex matrices are the L?wner and the minus partial orders. Motivated by applications in statistics we study these partial orders on H+ n (R). We describe the form of all surjective maps on H+ n (R), n > 1, that preserve the L?wner partial order in both directions. We present an equivalent definition of the minus partial order on H+ n (R) and also characterize all surjective, additive maps on H+ n (R), n ? 3, that preserve the minus partial order in both directions.

scholarly journals Preservers of partial orders on the set of all variance-covariance matrices

Filomat ◽  
2020 ◽  
Vol 34 (9) ◽  
pp. 3015-3030
Author(s):  
Iva Golubic ◽  
Janko Marovt
Keyword(s):  
Partial Order ◽  
Positive Semidefinite ◽  
Covariance Matrices ◽  
Partial Orders ◽  
Equivalent Definition ◽  
Definition Of ◽  
Square Complex ◽  
Additive Maps ◽  
Minus Partial Order ◽  
Real Matrices

Let H+n(R) be the cone of all positive semidefinite n x n real matrices. Two of the best known partial orders that were mostly studied on subsets of square complex matrices are the L?wner and the minus partial orders. Motivated by applications in statistics we study these partial orders on H+n(R). We describe the form of all surjective maps on H+ n (R), n > 1, that preserve the L?wner partial order in both directions. We present an equivalent definition of the minus partial order on H+n(R) and also characterize all surjective, additive maps on H+ n (R), n ? 3, that preserve the minus partial order in both directions.

Monotone transformations on the cone of all positive semidefinite real matrices

2020 ◽  
Vol 70 (3) ◽  
pp. 733-744
Author(s):  
Iva Golubić ◽  
Janko Marovt
Keyword(s):  
Partial Order ◽  
Covariance Matrix ◽  
Applied Mathematics ◽  
Positive Semidefinite ◽  
Partial Orders ◽  
Semidefinite Matrix ◽  
Additive Maps ◽  
Minus Partial Order ◽  
Real Matrices ◽  
Variance Covariance Matrix

AbstractLet $\begin{array}{} \displaystyle H_{n}^{+} \end{array}$(ℝ) be the cone of all positive semidefinite (symmetric) n × n real matrices. Matrices from $\begin{array}{} \displaystyle H_{n}^{+} \end{array}$(ℝ) play an important role in many areas of engineering, applied mathematics, and statistics, e.g. every variance-covariance matrix is known to be positive semidefinite and every real positive semidefinite matrix is a variance-covariance matrix of some multivariate distribution. Three of the best known partial orders that were mostly studied on various sets of matrices are the Löwner, the minus, and the star partial orders. Motivated by applications in statistics authors have recently investigated the form of maps on $\begin{array}{} \displaystyle H_{n}^{+} \end{array}$(ℝ) that preserve either the Löwner or the minus partial order in both directions. In this paper we continue with the study of preservers of partial orders on $\begin{array}{} \displaystyle H_{n}^{+} \end{array}$(ℝ). We characterize surjective, additive maps on $\begin{array}{} \displaystyle H_{n}^{+} \end{array}$(ℝ), n ≥ 3, that preserve the star partial order in both directions. We also investigate the form of surjective maps on the set of all symmetric real n × n matrices that preserve the Löwner partial order in both directions.

WEIGHTED CORE–EP INVERSE AND WEIGHTED CORE–EP PRE-ORDERS IN A -ALGEBRA

2019 ◽  
pp. 1-35 ◽  
Author(s):  
DIJANA MOSIĆ
Keyword(s):  
Partial Order ◽  
Hilbert Spaces ◽  
Partial Orders ◽  
Linear Operators ◽  
Bounded Linear Operators ◽  
Bounded Linear ◽  
The Core ◽  
Invertible Elements ◽  
Minus Partial Order

We define extensions of the weighted core–EP inverse and weighted core–EP pre-orders of bounded linear operators on Hilbert spaces to elements of a $C^{\ast }$ -algebra. Some properties of the weighted core–EP inverse and weighted core–EP pre-orders are generalized and some new ones are proved. Using the weighted element, the weighted core–EP pre-order, the minus partial order and the star partial order of certain elements, new weighted pre-orders are presented on the set of all $wg$ -Drazin invertible elements of a $C^{\ast }$ -algebra. Applying these results, we introduce and characterize new partial orders which extend the core–EP pre-order to a partial order.

Some Spectral Properties of Polar Decompositions

1984 ◽  
Vol 36 (6) ◽  
pp. 973-985
Author(s):  
Bryan E. Cain
Keyword(s):  
Polar Decomposition ◽  
Positive Semidefinite ◽  
Unitary Matrix ◽  
List Type ◽  
Complex Matrix ◽  
Important Special Case ◽  
Definition Of ◽  
Image Position ◽  
Square Complex ◽  
Special Case

The results in this paper respond to two rather natural questions about a polar decomposition A = UP, where U is a unitary matrix and P is positive semidefinite. Let λ1, …, λn be the eigenvalues of A. The questions are:(A) When will |λ1|, …, |λn| be the eigenvalues of P?(B) When will λ1/|λ1|, …, λn/|λn| be the eigenvalues of U?The complete answer to (A) is “if and only if U and P commute.” In an important special case the answer to (B) is “if and only if U2 and P commute.“Since these matters are best couched in terms of two different inertias, we begin with a unifying definition of inertia which views all inertias from a single perspective.For each square complex matrix A and each complex number z let m(A, z) denote the multiplicity of z as a root of the characteristic polynomial

THE NATURAL PARTIAL ORDER ON LINEAR SEMIGROUPS WITH NULLITY AND CO-RANK BOUNDED BELOW

2014 ◽  
Vol 91 (1) ◽  
pp. 104-115 ◽  
Author(s):  
SUREEPORN CHAOPRAKNOI ◽  
TEERAPHONG PHONGPATTANACHAROEN ◽  
PONGSAN PRAKITSRI
Keyword(s):  
Semigroup Forum ◽  
Partial Order ◽  
Lower Bounds ◽  
Partial Orders ◽  
Natural Partial Order ◽  
Linear Transformations ◽  
Bounded Below

AbstractHiggins [‘The Mitsch order on a semigroup’, Semigroup Forum 49 (1994), 261–266] showed that the natural partial orders on a semigroup and its regular subsemigroups coincide. This is why we are interested in the study of the natural partial order on nonregular semigroups. Of particular interest are the nonregular semigroups of linear transformations with lower bounds on the nullity or the co-rank. In this paper, we determine when they exist, characterise the natural partial order on these nonregular semigroups and consider questions of compatibility, minimality and maximality. In addition, we provide many examples associated with our results.

Remarks on some generalization of the notion of microscopic sets

2020 ◽  
Vol 70 (6) ◽  
pp. 1349-1356
Author(s):  
Aleksandra Karasińska
Keyword(s):  
General Sense ◽  
Equivalent Definition ◽  
Definition Of ◽  
Null Set

AbstractWe consider properties of defined earlier families of sets which are microscopic (small) in some sense. An equivalent definition of considered families is given, which is helpful in simplifying a proof of the fact that each Lebesgue null set belongs to one of these families. It is shown that families of sets microscopic in more general sense have properties analogous to the properties of the σ-ideal of classic microscopic sets.

Inverse complements and strongly unit regular elements

2021 ◽  
pp. 2250190
Author(s):  
Umashankara Kelathaya ◽  
Savitha Varkady ◽  
Manjunatha Prasad Karantha
Keyword(s):  
Partial Order ◽  
Regular Element ◽  
Generalized Inverse ◽  
Generalized Inverses ◽  
Order Unit ◽  
Regular Elements ◽  
Strongly Regular ◽  
Minus Partial Order

In this paper, the notion of “strongly unit regular element”, for which every reflexive generalized inverse is associated with an inverse complement, is introduced. Noting that every strongly unit regular element is unit regular, some characterizations of unit regular elements are obtained in terms of inverse complements and with the help of minus partial order. Unit generalized inverses of given unit regular element are characterized as sum of reflexive generalized inverses and the generators of its annihilators. Surprisingly, it has been observed that the class of strongly regular elements and unit regular elements are the same. Also, several classes of generalized inverses are characterized in terms of inverse complements.

Partial orders on the power sets of Baer rings

2019 ◽  
Vol 19 (01) ◽  
pp. 2050011 ◽  
Author(s):  
B. Ungor ◽  
S. Halicioglu ◽  
A. Harmanci ◽  
J. Marovt
Keyword(s):  
Partial Order ◽  
Partial Orders ◽  
Linear Operators ◽  
Bounded Linear ◽  
Von Neumann ◽  
Baer Ring ◽  
Power Set ◽  
Von Neumann Regular ◽  
Baer Rings ◽  
The Right

Let [Formula: see text] be a ring. Motivated by a generalization of a well-known minus partial order to Rickart rings, we introduce a new relation on the power set [Formula: see text] of [Formula: see text] and show that this relation, which we call “the minus order on [Formula: see text]”, is a partial order when [Formula: see text] is a Baer ring. We similarly introduce and study properties of the star, the left-star, and the right-star partial orders on the power sets of Baer ∗-rings. We show that some ideals generated by projections of a von Neumann regular and Baer ∗-ring [Formula: see text] form a lattice with respect to the star partial order on [Formula: see text]. As a particular case, we present characterizations of these orders on the power set of [Formula: see text], the algebra of all bounded linear operators on a Hilbert space [Formula: see text].

scholarly journals A Study of Completely Inverse Paramedial AG-Groupoids

2021 ◽  
pp. 101-115
Author(s):  
Muhammad Rashad ◽  
Imtiaz Ahmad ◽  
Faruk Karaaslan
Keyword(s):  
Partial Order ◽  
Partial Orders ◽  
Left Identity ◽  
Normal Congruence ◽  
Paramedial Law ◽  
Congruence Pair

A magma S that meets the identity, xy·z = zy·x, ∀x, y, z ∈ S is called an AG-groupoid. An AG-groupoid S gratifying the paramedial law: uv · wx = xv · wu, ∀ u, v, w, x ∈ S is called a paramedial AGgroupoid. Every AG-grouoid with a left identity is paramedial. We extend the concept of inverse AG-groupoid [4, 7] to paramedial AG-groupoid and investigate various of its properties. We prove that inverses of elements in an inverse paramedial AG-groupoid are unique. Further, we initiate and investigate the notions of congruences, partial order and compatible partial orders for inverse paramedial AG-groupoid and strengthen this idea further to a completely inverse paramedial AG-groupoid. Furthermore, we introduce and characterize some congruences on completely inverse paramedial AG-groupoids and introduce and characterize the concept of separative and completely separative ordered, normal sub-groupoid, pseudo normal congruence pair, and normal congruence pair for the class of completely inverse paramedial AG-groupoids. We also provide a variety of examples and counterexamples for justification of the produced results.

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