In this paper, we investigate the determinants involving some trigonometric functions. We establish a connection between these determinants and the special values of Dirichlet L-functions, thereby extending Guo's results (Determinants of trigonometric functions and class numbers. Linear Algebra Appl. 2022;653:33–43) to arbitrary positive integers n. In addition, we also prove a conjecture raised by Zhi-Wei Sun. Our main tool is the spectral decomposition of some linear operators. By the same method we obtain an explicit formula for the determinants of sine matrices. This formula is expressed as a product of Gauss sums attached to Dirichlet characters.

This paper is dedicated to the research of new classes of matrices defined through products involving the power A k of a square complex matrix A, and its core–EP inverse A ◯ † . Motivated by the identities A A ◯ † A l = A l and A ◯ † A l + 1 = A l for ind ( A ) ≤ l , we examine the matrix products A k A ◯ † A and A k + 1 A ◯ † , where k ∈ N , as well as A A ◯ † A k and A ◯ † A k + 1 , for k ∈ N and k < ind ( A ) . The matrices A k A ◯ † A and A k + 1 A ◯ † , for k ∈ N , represent two new types of k-CEP matrices, each serving as both an inner and outer inverse of ( A D ) k and ( A ◯ † ) k , respectively. We establish several fundamental properties, characterizations, and expressions for these matrices. In addition, the matrices A A ◯ † A k and A ◯ † A k + 1 , with k ∈ N and k < ind ( A ) , are two new types of CEP-k matrices. Consequently, four new classes of square matrices are introduced. Finally, the proposed k-CEP matrices are shown to be effective tools for solving certain linear equations, thereby highlighting their practical relevance and potential for application.

Barik et al. [On nonsingular trees and a reciprocal eigenvalue property. Linear Multilinear Algebra, 54(6)(2006) 453-465] provided a combinatorial description of the inverse of the adjacency matrix for bipartite graphs with a unique perfect matching in terms of mm-alternating (matching-matching alternating) paths, enabling deeper exploration of bipartite graph properties. However, for non-bipartite graphs with a unique perfect matching, a similar framework remained an open challenge. In 2022, a combinatorial description was established for non-bipartite unicyclic graphs with a unique perfect matching. This paper enhances the comprehension of non-bipartite graphs by offering a detailed combinatorial characterization of the inverse of the adjacency matrix for non-bipartite bicyclic graphs with a unique perfect matching, employing mm-alternating paths between vertices. It also identifies the conditions that make the adjacency matrix of a bicyclic graph unimodular and establishes criteria for determining whether the inverse of the adjacency matrix of a non-bipartite bicyclic graph is signature similar to a 0-1 matrix.

In this paper, we consider a spectral analysis of the Correlated Random Walk (CRW) on the path. We apply an analytical method for the Quantum Walk to CRW. For the isospectral coin cases, we obtain all of the eigenvalues and the corresponding eigenvectors of the time evolution operator of CRW. We show that most of them are described by the eigenvalues and the eigenvectors of the Jacobi matrix related to the random walks on the path (the birth and death chains). The remaining eigenvector corresponding to an eigenvalue 1, is characterized by the stationary measure of the same birth and death chain. As a consequence, we derive the limiting distribution.

We extend the classical Perron–Frobenius theory to tensors that may have negative entries, thereby broadening the scope of spectral analysis beyond the nonnegative setting. Under certain sufficient conditions, we establish the existence of a Perron–Frobenius eigenpair for such tensors and characterize the corresponding spectral radius. As an application, we propose a novel concept termed Perron–Frobenius splitting for tensors, which facilitates the solution of multi-linear systems via tensor splitting iterative methods. This framework generalizes the regular and weak regular splittings of the tensors. Furthermore, we provide convergence analyses and comparison theorems for the Perron–Frobenius splittings of the tensors.

In this work, we introduce a new type of generalized inverse called dual core-EP generalized inverse (in short, DCEPGI) for dual square matrices. We analyse the existence and uniqueness of the DCEPGI inverse and its compact formula using dual Drazin and dual MP inverse. Moreover, some characterizations using core-EP decomposition are obtained. We present a new dual matrix decomposition named the dual core-EP decomposition for square dual matrices. In addition, some relationships with other dual generalized inverses are established. As an application, solutions to system of linear dual equations are derived.

The power of orthogonality over the real and complex numbers lies in its use in computational and numerical methods. In this article, we discuss two different quasi-inner products over the finite fields, provide illustrative examples and develop additional results on how self-orthogonal vectors affect properties of these quasi-inner product spaces. We point out an error in a published paper and provide a correction. We examine relationships between a subspace W and its orthogonal complement. We then discuss different types of bases for subspaces and their implications. Finally, we compare and contrast the properties of quasi-inner product spaces depending on whether the transpose or the conjugate transpose is used to define the quasi-inner product space.

Computational methods that guarantee accurate solutions to linear algebra problems are of great interest in many applied contexts. These scenarios often involve particular matrix families that can benefit from a tailored analysis. In this work, we study a recently introduced class of structured matrices termed geometric r-Frank matrices, which are a one-parameter generalization of their classical version. Explicit bidiagonal factorizations for these matrices are derived, providing necessary and sufficient conditions for their total positivity. As a consequence, all eigenvalues and singular values can be determined with excellent relative accuracy under mild assumptions. Furthermore, we carry out a perturbation analysis for the bidiagonal factors and the determinants, establishing structured condition numbers that depend on the relative gaps of the underlying data. In addition, we develop efficient algorithms to compute the determinant of geometric r-Frank matrices together with running absolute and relative error bounds. Numerical experiments demonstrate the effectiveness and reliability of the proposed methods, even under challenging conditions.

The limiting probability distribution is one of the key characteristics of a Markov chain since it shows its long-term behaviour. In this paper, for a higher order Markov chain, we establish some properties related to its exact limiting probability distribution, including a sufficient condition for the existence of such a distribution. Our results extend the corresponding conclusions on first order chains. Besides, they complement the existing results concerning higher order chains which rely on approximation schemes or two-phase power iterations. Several illustrative example are also given.

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.