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Recently, Ferreyra and Malik (Core and strongly core orthogonal matrices. Linear Multilinear Algebra. 2021;1–16. doi:) have proved that if A is strongly core orthogonal to B, then rk (A + B) = rk (A) + rk (B) and (A + B) ◯ # = A ◯ # + B ◯ # . But whether the reverse holds is an open question. In this paper, we solve the problem completely and get some new characterizations of strong core orthogonality. [ABSTRACT FROM AUTHOR]

In this paper, we introduce the m -core inverse in the Minkowski space, and get a sufficient and necessary condition for the existence of the inverse and some other related properties. Furthermore, by using the inverse, we introduce the m -core partial ordering and obtain solutions (or restricted least squares solutions) of some matrix equations in the Minkowski space. [ABSTRACT FROM AUTHOR]

In this article, we will present some results relating sharp ordering (a ≤ ♯ b) and investigate the properties of one-sided cyclic ideals in rings. Necessary and sufficient conditions for the invertibility and the group invertibility of the linear combination c 1 a + c 2 b are given, where a is below b under the sharp order, a and b are elements of an algebra, and c1, c2 are elements of an arbitrary field. [ABSTRACT FROM AUTHOR]

In this paper, we introduce the notion of the EP-nilpotent decomposition and present some of its applications. By applying the decomposition, we introduce two partial orders (the E-N partial order and the E-S partial order) and characterize their properties. The two partial orders above are non-minus-type partial orders, although the method used to introduce the two partial orders is similar to that of the C-N partial order, a minus-type partial order. [ABSTRACT FROM AUTHOR]

In this paper, we use the well-known Löwner order and the core partial order to introduce a new partial orderon the class of core matrices which is not dominated by any of the known matrix partial orders. We characterize, study its relations with the Löwner partial order under constraints, and exemplify its differences with other partial orders. [ABSTRACT FROM PUBLISHER]

In this paper, we derive an interesting result that the mixed-type reverse order lawshold if and only if the bounded linear operator productsis invariant, whereis taken, for instance, as,,,,as well as. Moreover, the invariance property ofis related to the reverse order law, and we point out that such an invariance property occurs if and only ifandare left orandare right invertible operators, oris left andis right invertible operator, where,andare nonzero regular operators. [ABSTRACT FROM PUBLISHER]

In this note, we revisit the core inverse and the core partial ordering introduced by Baksalary and Trenkler [Linear Multilinear Algebra. 2010;58:681–697]. We prove that the core inverse ofis the unique solution ofand, and establish several characterizations of the core inverse, the core partial ordering and the reverse order law for the core inverse. [ABSTRACT FROM PUBLISHER]

We find expressions for many types of generalized inverses of an arbitrary square complex matrix by using two representations given in [Benítez J. A new decomposition for square matrices. Electron. J. Linear Algebra. 2010;20:207–225] and in [Hartwig RE, Spindelböck K. Matrices for whichandcommute. Linear Multilinear Algebra. 1984;14:241–256]. [ABSTRACT FROM PUBLISHER]

Let ℛ be a ring and a, b∈ℛ satisfy aba = a and bab = b. We characterize when ab − ba is invertible. This study is specialized when ℛ has an involution and when b is the Moore–Penrose inverse of a. [ABSTRACT FROM PUBLISHER]

In this paper, we present the explicit expressions of the perturbation of the Drazin inverse under different conditions. Also, we give the upper bounds of ‖(A+E)D−A D‖ P /‖A D‖ P for these cases. [ABSTRACT FROM PUBLISHER]

We consider the various representations of the outer inverse of an operator A on Banach spaces. [ABSTRACT FROM PUBLISHER]

Let T 1 and T 2 be two n × n tripotent matrices and c 1, c 2 two nonzero complex numbers. We mainly study the nonsingularity of combinations T = c 1 T 1 + c 2 T 2 − c 3 T 1 T 2 of two tripotent matrices T 1 and T 2, and give some formulae for the inverse of c 1 T 1 + c 2 T 2 − c 3 T 1 T 2 under some conditions. Some of these results are given in terms of group invertible matrices. [ABSTRACT FROM AUTHOR]

Let X † denotes the Moore--Penrose pseudoinverse of a matrix X. We study a number of situations when (aA + bB)† = aA + bB provided a, b ∈ ℂ ∖ {0} and A, B are n × n complex matrices such that A † = A and B † = B. [ABSTRACT FROM PUBLISHER]

We derive a very short expression for the group inverse of a1 + ··· + an when a1, ... , an are elements in an algebra having group inverse and satisfying aiaj = 0 for i < j. We apply this formula in order to find the group inverse of 2 × 2 block operators under some conditions. [ABSTRACT FROM AUTHOR]

In this article, we discuss the group inverse of aP + bQ + cPQ + dQP + ePQP + fQPQ + gPQPQ of idempotent matrices P and Q, where a, b, c, d, e, f, g ∈  and a ≠ 0, b ≠ 0, put forward its explicit expressions, and some necessary and sufficient conditions for the existence of the group inverse of aP + bQ + cPQ. [ABSTRACT FROM AUTHOR]

An n × n complex matrix A is said to be k-potent if Ak = A. Let T1 and T2 be k-potent and c1 and c2 be two nonzero complex numbers. We study the range space, null space, nonsingularity and group invertibility of linear combinations T = c1T1 + c2T2 of two k-potent matrices T1 and T2. [ABSTRACT FROM AUTHOR]