Inverse complements and strongly unit regular elements

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.

Preservers of partial orders on the set of all variance-covariance 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.

On some orders in ∗-rings based on the core-EP decomposition

We study certain relations in unital rings with involution that are derived from the core-EP decomposition. The notion of the WG pre-order and the C-E partial order is extended from [Formula: see text], the set of all [Formula: see text] matrices over [Formula: see text], to the set [Formula: see text] of all core-EP invertible elements in an arbitrary unital ring [Formula: see text] with involution. A new partial order is introduced on [Formula: see text] by combining the WG pre-order and the well known minus partial order, and a new characterization of the core-EP pre-order in unital proper ∗-rings is presented. Properties of these relations are investigated and some known results are thus generalized.

Inverse complements of a matrix and applications

In this paper, the concept of “Inverse Complemented Matrix Method”, introduced by Eagambaram (2018), has been reestablished with the help of minus partial order and several new properties of complementary matrices and the inverse of complemented matrix are discovered. Class of generalized inverses and outer inverses of given matrix are characterized by identifying appropriate inverse complement. Further, in continuation, we provide a condition equivalent to the regularity condition for a matrix to have unique shorted matrix in terms of inverse complemented matrix. Also, an expression for shorted matrix in terms of inverse complemented matrix is given.

Monotone transformations on the cone of all positive semidefinite real matrices

Let $\begin{array}{} \displaystyle H_{n}^{+} \end{array}$(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.

Preservers of partial orders on the set of all variance-covariance 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.

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

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.