A well-known characterization of Jordan vectors of a matrix polynomial $L(z)$ is generalized to a characterization of Jordan vectors of the operator-valued function $Q(z)$ at an eigenvalue $\alpha \in \mathbb{C}$. The results are then applied to solve a system of nonlinear ordinary differential equations.

On the inversion of polynomials of discrete Laplace matrices

Circumferential guided wave detection technology can serve as an alternative method for detecting casing bond defects. Due to the presence of the cement cladding, the circumferential SH guided waves transmit shear waves into the cement cladding as they propagate in the cementing casing, which cause the circumferential SH guided waves to show attenuation characteristics. In this study, the cementing casing structure was considered as a steel substratum semi-infinite domain cemented cladding pipe structure, and the corresponding dispersion and attenuation characteristics of circumferential SH guided waves were numerically solved based on the state matrix and Legendre polynomial hybrid method. In addition, a finite element simulation model of cementing casing was established to explore the interaction between SH guided waves and bonding defects. The relationship between the amplitude of SH guided waves and the size of the bonding defects was established through the attenuation coefficient. Moreover, an experimental platform for cementing casing detection is constructed to detect bonding defects of different sizes and to achieve the acoustic analysis of cementing defects in cementing casing, which provides a research path for the non-destructive testing and evaluation of bonding defects in cementing casing.

Generator polynomial matrices of the Galois hulls of multi-twisted codes

Theory of Smith Forms for Bivariate Polynomial Matrices

Moments of Characteristic Polynomials of Non-symmetric Random Matrices

Fluctuations of the Stieltjes transform of the empirical spectral distribution of selfadjoint polynomials in Wigner and deterministic diagonal matrices

Nonlinear eigenvalue solver for spectral element of beam structures: An exponential matrix polynomial approximation with weighted residual method

Umbral study of Mittag-Leffler-Laguerre matrix polynomials

This paper introduces a digital adaptive control framework for large-scale multivariable systems, integrating matrix linear Diophantine equations with block pole placement. The main innovation lies in adaptively relocating the full eigenstructure using matrix polynomial representations and a recursive identification algorithm for real-time parameter estimation. The proposed method achieves accurate eigenvalue placement, strong disturbance rejection, and fast regulation under model uncertainty. Its effectiveness is demonstrated through simulations on a large-scale winding process, showing precise tracking, low steady-state error, and robust decoupling. Compared with traditional non-adaptive designs, the approach ensures superior performance against parameter variations and noise, highlighting its potential for high-performance industrial applications.

In this paper, we explicitly provide expressions for a sequence of orthogonal polynomials associated with a weight matrix of size N , constructed from a collection of scalar weights w_{1}, \ldots, w_{N} of the form W(x) = T(x)\operatorname{diag}(w_{1}(x), \ldots, w_{N}(x))T(x)^{\ast} , where T(x) is a specific polynomial matrix. We provide sufficient conditions on the scalar weights to ensure that the weight matrix W is irreducible. Furthermore, we give sufficient conditions on the scalar weights to ensure that each term in the constructed sequence of matrix orthogonal polynomials is an eigenfunction of a differential operator. We also study the Darboux transformations and bispectrality of the orthogonal polynomials in the particular case where the scalar weights are the classical weights of Jacobi, Hermite, and Laguerre. With these results, we construct a wide variety of bispectral matrix-valued orthogonal polynomials of arbitrary size, which satisfy a second-order differential equation.

Let G be a connected threshold graph. We derive a concise formula to find the eigenvalues of the distance Laplacian matrix of G and show that this formula characterizes all connected threshold graphs. Additionally, by this formula, we give a simple procedure to determine the number of dominant and pendant vertices of G. We then classify all the dominant vertices of G from the eigenvectors of the distance Laplacian matrix. Finally, we prove that, among all connected threshold graphs, the coefficients of the characteristic polynomial of the distance Laplacian matrix attain the minimum and maximum values at the complete graph and the star tree, respectively.

Rectangular multiparameter eigenvalue problems (RMEP) consisting of a single multivariate polynomial have received interest among the researchers due to their applications in diverse scientific domain, particularly in optimal least square model problems. A common method for determining the optimal least squares of linear time-invariant dynamical systems (LTI) and autoregressive moving average (ARMA) models are obtained from the solution of the rectangular polynomial two-parameter eigenvalue problems (RPTEP). This makes it necessary to find effective solution methods for this particular kind of eigenvalue problem. Linearizing the matrix polynomial associated with RMEP followed by the conversion to a known form of linear two-parameter eigenvalue problem, and then using the Vandermonde compression is the currently available method in the literature. In this paper, we present a two-parameter matrix pencil method to obtain the solution of the RPTEP, that can be used as a ready reference to compute the solution of LTI and ARMA. Numerical works are performed to verify the computational efficiency of the method.

Row completion of polynomial and rational matrices

A modified fixed-point iteration method for a class of polynomial matrix equations

We obtain a tail bound for the least non-zero singular value of A − z when A is a random matrix and z is an eigenvalue of A in a neighborhood of a given point z0 in the bulk of the spectrum. The argument relies on a resolvent comparison and a tail bound for Gauss-divisible matrices. The latter can be obtained by the method of partial Schur decomposition. Using this bound we prove that any finite collection of components of a right eigenvector corresponding to an eigenvalue uniformly sampled from a neighborhood of a point in the bulk is Gaussian. A byproduct of the calculation is an asymptotic formula for the odd moments of the absolute value of the characteristic polynomial of real Gauss-divisible matrices.

For a finite group G, the coprime graph ΓG of G is defined as the graph with vertex set G, the group itself, and two distinct vertices u, v in ΓG are adjacent if and only if gcd(|u|, |v|) = 1, where |u| is the order of u. This study analyzes the characteristic polynomial of matrices for the dihedral group of order 2n, where n is a power of a prime number. In addition, this paper examines the characteristic polynomial of the matrices for a power of a prime number n. The energy of the graph is also obtained.

On the Divisibility with Remainder of Polynomial Matrices Over an Arbitrary Field

On the Triangular Form of a Polynomial Matrix of Simple Structure and Its Invariants with Respect to Semiscalar Equivalence

On the precise deviations of the characteristic polynomial of a random matrix