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The solvability of several new constrained quaternion matrix equations is investigated, and their unique solutions are presented in terms of the weighted MPD inverse and weighted DMP inverse of suitable matrices. It is interesting to consider some exceptional cases of these new equations and corresponding solutions. Determinantal representations for the solutions of the equations as mentioned earlier are established as sums of appropriate minors. In order to illustrate the obtained results, a numerical example is shown.

The aim of this paper is to establish formulae for the Drazin inverse of anti-triangular block matrices under new assumptions in literature. Precisely, we consider the Drazin inverse of three kinds of anti-triangular block matrices. Applying these results, we present new expressions for the Drazin inverse of an arbitrary block matrix. In this way, we generalize a list of earlier results and illustrate it with three examples.

Let [Formula: see text] be a unital ring with involution. The notions of 1MP-inverse and MP1-inverse are 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 Moore–Penrose invertible elements in [Formula: see text]. We study partial orders on [Formula: see text] that are induced by 1MP-inverses and MP1-inverses. We also extend to the setting of Rickart ∗-rings the concept of another partial order, called the plus order, which has been recently introduced on the set of all bounded linear operators between Hilbert spaces. Properties of these relations are investigated and some known results are thus generalized.

To solve certain type of matrix equations and extend the notions of the Drazin-star and star-Drazin matrices, we define two new classes of rectangular matrices called the outer-star and star-outer matrices. We characterize these new matrices by algebraic and geometrical point of view and give properties of projectors determined by them. Various representations of outer-star and star-outer matrices as integral representations and limit representations are derived. We establish relations between outer-star and star-outer matrices with appropriate nonsingular matrices. Possible applications of outer-star and star-outer matrices in solving some systems of linear equations are presented.

In this paper, we present several equivalent conditions related to the forward order law for the core inverse of two matrices, i.e., $$(AB)^{{\tiny {\textcircled {\tiny \#}}}}=A^{{\tiny {\textcircled {\tiny \#}}}}B^{{\tiny {\textcircled {\tiny \#}}}}$$ . In addition, we consider problems when $$(AB)^{\dagger }=A^{{\tiny {\textcircled {\tiny \#}}}}B^{{\tiny {\textcircled {\tiny \#}}}}$$ , $$(AB)^{\#}=A^{{\tiny {\textcircled {\tiny \#}}}}B^{{\tiny {\textcircled {\tiny \#}}}}$$ and $$(AB)_{{\tiny {\textcircled {\tiny \#}}}}=A^{{\tiny {\textcircled {\tiny \#}}}}B^{{\tiny {\textcircled {\tiny \#}}}}$$ , respectively. Thus, we study some hybrid forward order laws too.

As solutions of some operator equations, we present two new outer inverses for bounded linear operators between two Hilbert spaces. Precisely, we introduce and characterize the weighted weak core i...

The class of polynomially normal operators is a wider class than the class of all normal operators. Inspired by some interesting well known facts about normal operators and by some recent work, we ...

Using compositions of the weighted core–EP inverse and Moore–Penrose inverse, we solve new systems of operator equations and define three new weighted generalized inverses of a bounded linear operator between two Hilbert spaces, which are called the W-MPCEP inverse, W-CEPMP inverse and W-MPCEPMP inverse. The W-MPCEP inverse extends the notion of the MPCEP inverse for an operator. We give operator matrix forms and some characterizations of W-MPCEP, W-CEPMP and W-MPCEPMP inverses. We also establish maximal classes of operators for which the expressions of W-MPCEP, W-CEPMP and W-MPCEPMP inverses are valid.

Our contribution is the development of novel representations and investigations of main properties of the MPCEP inverse. Precisely, we present representations of the MPCEP inverse which involve appropriate Moore–Penrose inverses, projections and full-rank decompositions, as well as limit and integral representations. Determinantal representations for the MPCEP inverse are also established. We study perturbation formulae with perturbation bounds of MPCEP inverse. An equivalent condition for the continuity of the MPCEP inverse is proposed. The MPCEP inverse of a suitable upper block triangular matrix is given. The successive matrix squaring algorithm and splitting method for computing the MPCEP inverse are presented. Some appropriate constrained systems of linear equations are solved applying the MPCEP inverse.

The main contribution of this paper is to present new generalized inverses as weaker versions of a G-outer inverse. In particular, we define and characterize left and right G-outer inverses of rect...

To extend the notation of inner inverses, we define weighted inner inverses of a rectangular matrix. In particular, we introduce a W -weighted (B, C)-inner inverse of A, for given matrices A, W, B,...

We present the weighted weak group inverse, which is a new generalized inverse of operators between two Hilbert spaces, and we extend the notation of the weighted weak group inverse for rectangular matrices. Some characterizations and representations of the weighted weak group inverse are investigated. We also apply these results to define and study the weak group inverse for a Hilbert space operator. Using the weak group inverse, we define and characterize various binary relations.

According to known definition in the literature, the W-weighted outer inverses of A is defined as the outer inverses of WAW. We introduce and study more general weighted outer inverses of rectangul...

The paper surveys, classifies and investigates theoretically and numerically main classes of line search methods for unconstrained optimization. Quasi-Newton (QN) and conjugate gradient (CG) methods are considered as representative classes of effective numerical methods for solving large-scale unconstrained optimization problems. In this paper, we investigate, classify and compare main QN and CG methods to present a global overview of scientific advances in this field. Some of the most recent trends in this field are presented. A number of numerical experiments is performed with the aim to give an experimental and natural answer regarding the numerical one another comparison of different QN and CG methods.

In this paper, we consider Moore-Penrose invertible, group invertible, and core invertible elements in rings with involution to characterize EP, generalized normal, generalized Hermitian elements and generalized partial isometries. As a consequence, we obtain new characterizations for elements in rings with involution to be normal and Hermitian elements.

We introduce the e–core–EP inverse and the core–EP inverse for elements of Banach algebras. Several characterizations and representations for these inverses are presented using idempotents and Draz...

We obtain two different expressions for the weighted core–EP inverse of the sum $$A+E$$ for a given operator A and a perturbation operator E, under different conditions on the perturbing operator. As a consequence of these results, we get some relations between idempotents determined by the weighted core–EP inverses of a given operator and its perturbation and we present perturbation bounds for the weighted core–EP inverse. The application to the solution of a perturbed equation is given. We also apply these results to obtain perturbation results for the core–EP inverse.

Various compositions of the Drazin inverse, the group inverse or the core-EP inverse with the Moore-Penrose inverse have investigated last years. Solving some type of matrix equations, we introduce...

Let R be a ring and a,b,c∈R. In this paper, it is shown that if a is left (b,c)-invertible and cab is regular, then a has at least one regular left (b,c)-inverse. And the expression of the (b,c)-in...

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.

Using the Moore-Penrose inverse and the core-EP inverse, we define a new generalized inverse (called the MPCEP inverse) for a Hilbert space operator. Several equivalent conditions for a Hilbert spa...

In this paper, we generalized Hua’s identity to the $$\{1, 2\}$$-inverse case. In addition, theconditions that make Hua’s identities true for the Moore–Penrose inverse, group inverse, core inverse and dual core inverse are given, respectively.

Let R be a unital ring with involution. The core inverses of differences and products of projections in R are investigated. Let $$p, q\in R$$ be two projections, we obtained equivalent conditions such that $$p-q$$, $$(p-q)^2$$, $$1-qp$$, $$pq+qp$$ and $$(pq-qp)^2$$ are core invertible, respectively. Moreover, expressions for their core inverses are given.

We define a weighted extended g-Drazin inverse for Banach algebra elements, generalizing the concepts of extended g-Drazin inverse and weighted g-Drazin inverse. We characterize weighted extended g-Drazin invertible elements and present some representations of a weighted extended g-Drazin inverse. Applying these results, we introduce and investigate a weighted extended Drazin inverse.

We introduce new classes of generalized Drazin inverses in a ring, which are called the generalized n-strong Drazin inverse and pseudo n-strong Drazin inverse for arbitrary n∈N. Some basic properti...

We develop maximal classes of operators for which the representations of the weighted gDMP inverse, the weighted core–EP inverse and the weighted CMP inverse are valid. Applying our results, we obt...

We introduce and investigate a new generalized inverse as an extension of the g-Drazin inverse in a Banach algebra. This new inverse will be called an extended g-Drazin inverse. Using idempotents, we characterize this inverse and give some its representations. Also, we prove generalizations of Cline’s formula for extended g-Drazin inverse. As a consequence of our results, we present the definition and characterizations of an extended Drazin inverse.

In this paper, we introduce a square hyponormal operator as a bounded linear operator T on a complex Hilbert space H such that T2 is a hyponormal operator, and we investigate some basic properties of this operator. Under the hypothesis ?(T) ? (-?(T)) ? {0}, we study spectral properties of a square hyponormal operator. In particular, we show that if z and w are distinct eigen-values of T and x,y ? H are corresponding eigen-vectors, respectively, then ?x,y? = 0. Also, we define nth hyponormal operators and present some properties of this kind of operators.