A kth partition reduction method is used to obtain new upper bounds for the inverses of H-matrices with their subclasses. The estimates are expressed via the determinants of kth order matrices, which is sharper than the existed ones. Numerical experiments with various random matrices show that they are stable and better than the estimates presented in literature. In addition, by reduce the order of the variables, the error bound of linear complementarity problems relies on less variables and reaches at the vertex on the reduced closed region. We use the upper bounds in order to improve known error estimates for linear complementarity problems with H-matrices.

This paper introduces and examines a new class of nuclear operators within the family of Lipschitz operators, referred to as Lipschitz σ-nuclear operators. We investigate their fundamental properties, particularly their characterization through factorization theorems. Furthermore, we establish a characterization of Lipschitz compact operators based on the injectivity of Lipschitz σ-nuclear operators and determine the dual space of these operators.

Let Φ ~ = ( G , w ) be a connected weighted graph and A ( Φ ~ ) be its adjacency matrix. The rank r ( G , w ) of Φ ~ is defined as the rank of its adjacency matrix A ( Φ ~ ) . For a given underlying graph G, the girth g ( G ) (or simply g) is the length of a shortest cycle (shortest graph-theoretic distance) in G. In this paper, we establish the bound r ( G , w ) ≥ g ( G ) − 2 for any weighted graph ( G , w ) . Furthermore, we characterize the cases where equality holds, specifically when r ( G , w ) = g − 2 and r ( G , w ) = g . Our results extend previous work on ordinary graphs by Zhou et al. (Linear Algebra Appl., 2021), signed graphs by Wu et al. (Linear Algebra Appl., 2022), and complex unit gain graphs by Khan (Linear Algebra Appl., 2024) to the setting of weighted graphs.

In this paper, we give necessary and sufficient conditions for a product of EP elements to be EP, in Dedekind-finite ∗ -rings. The proofs rely on descriptions of Morita contexts (via Peirce matrix forms) as corner rings of 2 × 2 matrix rings.

In this paper, we use the induced representation theory to derive some properties of the so called base matrices. As applications, we firstly present the weighted complexity of a connected arbitrary lift of a graph X by using the weighted complexity of X. Then we provide a general method to derive the properties of several L-functions of a graph or digraph by induced representation theory. In particular, several known results are derived as corollaries.

Let a, f be elements in a ring with pseudo core inverses a ◯ D , f ◯ D , and let b = f−a. We prove that the absorption law a ◯ D ( a + f ) f ◯ D = a ◯ D + f ◯ D holds if and only if 1 + a ◯ D b is invertible and the additive property f ◯ D = ( 1 + a ◯ D b ) − 1 a ◯ D is satisfied. We further characterize these properties and establish analogous results for other generalized inverses. Finally, we apply these results to the case of complex matrices.

This paper presents an analogue of the spectral mapping theorem for the pseudo S-spectra of quaternionic operators. The theorem is formulated using two set inclusions, and necessary and sufficient conditions for equality are identified. The standard mapping theorem appears as a special case of this theorem. A weaker version is also derived with potential applications in various scenarios. The pseudo S-spectra of rank-one and rank-one nilpotent quaternionic operators are explicitly characterized. Finally, the mapping theorem is further demonstrated for specific classes of quaternionic operators, including rank-one nilpotent, normal, self-adjoint, orthogonal projection, and left multiplication operators.

In this paper, we classify surjective maps that preserve order and spectrum on the positive cone and the real space of all self-adjoint elements in A W ∗ -algebras. Our results extend the work of P. Šemrl on von Neumann algebras of type I factors.

In a recent work (Linear Algebra Appl. 2017;520:215–241), the authors have proved that the Jacobi–Davidson iteration method attains cubic convergence rate locally for solving Hermitian standard eigenvalue problems. In this paper, we attempt to generalize these convergence results from Hermitian standard eigenvalue problems to Hermitian generalized eigenvalue problems. It indicates that the Jacobi–Davidson method at least converges linearly, and it converges quadratically and cubically if the involved correction equation is solved to the precision proportional to the norm of the current residual and to the precision proportional to the squares of the norm of the current residual, respectively. Numerical experiments are carried out to confirm these theoretical results.

For the dual generalized inverse, the dual index and dual core generalized inverses have just been introduced in Wang and Gao [The dual index and dual core generalized inverse. Open Math. 2023;21(1):Paper No. 20220592, 23]. The decomposition of dual core-EP generalized inverse for dual matrices has already been proposed recently in Sitha et al. [Dual core-EP generalized inverse and decomposition. Linear Multilinear Algebra. 2025. doi: 10.48550/arXiv.2502.04652]. In this study, a binary relation on dual core generalized inverse is introduced. We obtain characterizations of the dual core generalized inverses by applying the dual core-EP decomposition. A new dual binary core relation is introduced using a dual generalized core inverse. It is demonstrated that the D-core binary relation is a partial order relation, but in general, C-core relation is not a partial order.