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Purpose – The author studies forms over finite fields obtained as the determinant of Hermitian matrices and use these determinatal forms to define and study the base polynomial of a square matrix over a finite field. Design/methodology/approach – The authors give full proofs for the new results, quoting previous works by other authors in the proofs. In the introduction, the authors quoted related references. Findings – The authors get a few theorems, mainly describing some monic polynomial arising as a base polynomial of a square matrix. Originality/value – As far as the author knows, all the results are new, and the approach is also new.

In the field of radar target detection, the conventional approach is to employ the range profile energy accumulation method for detecting extended targets. However, this method becomes ineffective when dealing with non-stationary and non-uniform radar clutter scenarios, as well as long-distance targets with weak radar cross sections (RCSs). In such cases, the signal-to-noise ratio (SNR) of the target echo is severely degraded, rendering the energy accumulation detection algorithm unreliable. To address this issue, this paper presents a new extended target detection method based on the maximum eigenvalue of the Hermitian matrix. This method utilizes a detection model that incorporates observed data and employs the likelihood ratio test (LRT) theory to derive the maximum eigenvalue detector at low SNR. Specifically, the detector constructs a matrix using a sliding window block with the available data and then computes the maximum eigenvalue of the covariance matrix. Subsequently, the maximum eigenvalue matrix is transformed into a one-dimensional eigenvalue image, enabling extended target detection through analogy with the energy accumulation detection method. Furthermore, this paper analyzes the proposed extended target detection method from both theoretical and experimental perspectives, validating it through field-measured data. The results obtained from the measured data demonstrate that the method effectively enhances the SNR in low SNR conditions, thereby improving target detection performance. Additionally, the method exhibits robustness across different scattering center targets.

Batched matrix computing problems are widely existed in scientific computing and engineering applications.With rapid performance improvements,GPU has become an important tool to solve such problems.The eigenvalue decomposition belongs to the two-sided decomposition and must be solved by the iterative algorithm.Iterative numbers for different matrices can be varied.Therefore,designing eigenvalue decomposition algorithms for batched matrices on the GPU is more challenging than designing batched algorithms for the one-sided decomposition,such as LU decomposition.This paper proposes batched algorithms based on the Jacobi algorithms for eigenvalue decomposition of Hermitian matrices.For matrices that cannot reside in shared memory wholly,the block technique is used to improve the arithmetic intensity,thus improving the use of GPU resources.Algorithms presented in this paper run completely on the GPU,avoiding the communication between the CPU and GPU.Kernel fusion is adopted to decrease the overhead of launching kernel and global memory access.Experimental results on V100 GPU show that our algorithms are better than existing works.Performance evaluation results of the Roofline model indicate that our implementations are close to the upper bound,approaching 4.11TFLOPS.

This paper considers the Hermitian solutions of a new system of commutative quaternion matrix equations, where we establish both necessary and sufficient conditions for the existence of solutions. Furthermore, we derive an explicit general expression when it is solvable. In addition, we also provide the least squares Hermitian solution in cases where the system of matrix equations is not consistent. To illustrate our main findings, in this paper we present two numerical algorithms and examples.

For any simple connected graph G of order n, having eigen spectrum μ 1 ≥ μ 2 ≥ ⋯ ≥ μ n with middle eigenvalues μ H and μ L, where H = ⌊(n + 1)/2⌋ and L = ⌈(n + 1)/2⌉, the HOMO–LUMO gap is defined as as ΔG = μ H = μ L. In this article, a simple upper bound for the HOMO–LUMO gap corresponding to a special class of connected bipartite graphs is estimated. As an application, the upper bounds for the HOMO–LUMO gap of certain classes of nanotubes and nanotori are estimated.

We introduce the concept of the almost-companion matrix (ACM) by relaxing the non-derogatory property of the standard companion matrix (CM). That is, we define an ACM as a matrix whose characteristic polynomial coincides with a given monic and generally complex polynomial. The greater flexibility inherent in the ACM concept, compared to CM, allows the construction of ACMs that have convenient matrix structures satisfying desired additional conditions, compatibly with specific properties of the polynomial coefficients. We demonstrate the construction of Hermitian and unitary ACMs starting from appropriate third-degree polynomials, with implications for their use in physical-mathematical problems, such as the parameterization of the Hamiltonian, density, or evolution matrix of a qutrit. We show that the ACM provides a means of identifying the properties of a given polynomial and finding its roots. For example, we describe the ACM-based solution of cubic complex algebraic equations without resorting to the use of the Cardano-Dal Ferro formulas. We also show the necessary and sufficient conditions on the coefficients of a polynomial for it to represent the characteristic polynomial of a unitary ACM. The presented approach can be generalized to complex polynomials of higher degrees.

A novel relation between the Moore-Penrose inverses of two nullity-1 n × n Hermitian matrices which share a common null eigenvector is established, and its application in electrical networks is illustrated by applying the result to Laplacian matrices of graphs.

Abstract In the paper, the authors establish some inequalities for generalized eigenvalues of perturbation problems on Hermitian matrices and modify shortcomings of some known inequalities for generalized eigenvalues in the related literature.

For solving large sparse non-Hermitian positive definite linear equations, Bai et al. proposed the Hermitian and skew-Hermitian splitting methods (HSS). They recently generalized this technique to the normal and skew-Hermitian splitting methods (NSS). In this paper, we present an accelerated normal and skew-Hermitian splitting methods (ANSS) which involve two parameters for the NSS iteration. We theoretically study the convergence properties of the ANSS method. Moreover, the contraction factor of the ANSS iteration is derived. Numerical examples illustrating the effectiveness of ANSS iteration are presented.

A majorization relating the singular values of an off-diagonal block of a Hermitian matrix and its eigenvalues is obtained. This basic majorization inequality implies various new and existing results.