Class Of Matrices Research Articles

On the surprising effectiveness of a simple matrix exponential derivative approximation, with application to global SARS-CoV-2

The continuous-time Markov chain (CTMC) is the mathematical workhorse of evolutionary biology. Learning CTMC model parameters using modern, gradient-based methods requires the derivative of the matrix exponential evaluated at the CTMC's infinitesimal generator (rate) matrix. Motivated by the derivative's extreme computational complexity as a function of state space cardinality, recent work demonstrates the surprising effectiveness of a naive, first-order approximation for a host of problems in computational biology. In response to this empirical success, we obtain rigorous deterministic and probabilistic bounds for the error accrued by the naive approximation and establish a "blessing of dimensionality" result that is universal for a large class of rate matrices with random entries. Finally, we apply the first-order approximation within surrogate-trajectory Hamiltonian Monte Carlo for the analysis of the early spread of Severe acute respiratory syndrome coronavirus 2 (SARS-CoV-2) across 44 geographic regions that comprise a state space of unprecedented dimensionality for unstructured (flexible) CTMC models within evolutionary biology.

THE FREQUENCY METHOD FOR OPTIMAL IDENTIFICATION OF CLOSE-LOOP SYSTEM ELEMENTS

Context. The article is devoted to overcoming the contradictions between the assumptions adopted in known methods of closedloop control system identification and the design and conditions of its operation. The article presents a new method of identifying the transfer functions matrix of a two-level closed-loop control system element, which functions under the conditions of multidimensional stationary centered random influences.
 Objective. The purpose of the study, the results of which are presented in this paper, is to extend the indirect identification method to the case of estimating one of the two-level closed-loop control system elements’ dynamics model based on passive experiment data.
 Method. To solve the optimal identification problem, a variational method for minimizing the quality functional on the class of fractional-rational transfer function matrices was used.
 Results. As a result of the research, the identification problem formulation was formalized, the rules for obtaining experimental information about the input and output signals were determined, the rules for identifying the transfer functions matrix of a two-level closed-loop control system element, which minimizes the sum of the variances of identification errors in the frequency domain, and the verification of these rules was carried out.
 Conclusions. Justified rules allow to correctly determine transfer functions matrices of the closed-loop systems selected element when fulfilling the defined list of conditions. The closed-loop systems control paths signals analysis proves the possibility of the effect of changing these signals statistical means, even under conditions of only centered stationary input influences actions on the system. Based on this, the further development of research can be aimed at overcoming such effects.

Drazin theta and theta Drazin matrices

<p style='text-indent:20px;'>In this paper, two new classes of matrices - Drazin - theta matrix and theta - Drazin matrix are introduced for a square matrix of index <inline-formula><tex-math id="M3">\begin{document}$ m $\end{document}</tex-math></inline-formula>. Whenever the index is equal to one, we get special case of matrices called Group - theta matrix and theta - Group matrix respectively. Several characterizations of these matrices, the integral representations, representation in limit form and the representation in terms of rank factorization are obtained. Also, the relationship of Drazin -theta and theta - Drazin matrices with other well known generalized inverses are investigated. By applying the concept of Drazin - theta matrix, general solutions of certain types of matrix equations are characterized here.</p>

A Class of Random Matrices

A Class of Random Matrices

АСИМПТОТИЧНІ ВЛАСТИВОСТІ ВИПАДКОВИХ МАТРИЦЬ

The work explores the asymptotic properties of large-dimensional stochastic matrices N under the condition of independence of matrix elements or rows (columns). An analysis of the main properties of eigenvalues of stochastic matrices is conducted. The work is dedicated to investigating the asymptotic characteristics of random matrices under the absence of the second moment and also considers the presence of "heavy tails" in the corresponding transitions in the adjacency matrices of the respective graph. The main result of the work is formulated in terms of the transition matrix of a discrete Markov chain and its eigenvalues. In proving the theorem, a non-degenerate Markov chain is considered, describing a mathematical model of random processes that do not change over time with known transition probabilities between states and a corresponding stochastic matrix P, one of whose eigenvalues is unity, and all eigenvalues of the stochastic matrix do not exceed it in absolute value. The proof of this fact follows from the Perron–Frobenius theorem, which concerns the properties of positive matrices and their eigenvalues. Thus, the theorem considered manages to expand the class of random matrices A for which convergence of eigenvalues of the matrix can be applied under the conditions imposed on the elements of the adjacency matrix. Moreover, the imposed conditions are relaxed compared to classical results, where the existence of a finite second moment for the elements of the adjacency matrix is required. This result generalizes both classical results for the normal distribution and similar results of other authors. The obtained result can be used in graph clustering problems to choose the optimal number of clusters, namely, they can be applied to determine the optimal number of clusters in a Grid system, complex networks, in investigations of the structure of the web space, etc.

Crouzeix's conjecture for classes of matrices

For a matrix A which satisfies Crouzeix's conjecture, we construct several classes of matrices from A for which the conjecture will also hold. We discover a new link between cyclicity and Crouzeix's conjecture, which shows that Crouzeix's Conjecture holds in full generality if and only if it holds for the differentiation operator on a class of analytic functions. We pose several open questions, which if proved, will prove Crouzeix's conjecture. We also begin an investigation into Crouzeix's conjecture for symmetric matrices and in the case of 3×3 matrices, we show Crouzeix's conjecture holds for symmetric matrices if and only if it holds for analytic truncated Toeplitz operators.

Bush-type Butson Hadamard matrices

Bush-type Butson Hadamard matrices are introduced. It is shown that a nonextendable set of mutually unbiased Butson Hadamard matrices is obtained by adding a specific Butson Hadamard matrix to a set of mutually unbiased Bush-type Butson Hadamard matrices. A class of symmetric Bush-type Butson Hadamard matrices over the group \(G\) of \(n\)-th roots of unity is introduced that is also valid over any subgroup of \(G\). The case of Bush-type Butson Hadamard matrices of even order will be discussed.

Index-MP and MP-index matrices

To extend the concepts of the DMP, MPD and CMP inverses and solve several types of systems of equations, we define three new kinds of square matrices, called the index-MP, MP-index and MP-index-MP matrices. We prove several characterizations for new classes of matrices and their relations with significant generalized inverses. Several types of representations for these new matrices are established. We define new binary relations applying index-MP, MP-index and MP-index-MP matrices and we show that these relations are partial orders on the set including square matrices of index one.

Thermodynamic Formalism for Continuous-Time Quantum Markov Semigroups: the Detailed Balance Condition, Entropy, Pressure and Equilibrium Quantum Processes

Let [Formula: see text] denote the set of [Formula: see text] by [Formula: see text] complex matrices. Consider continuous time quantum semigroups [Formula: see text], [Formula: see text], where [Formula: see text] is the infinitesimal generator. If we assume that [Formula: see text], we will call [Formula: see text], [Formula: see text] a quantum Markov semigroup. Given a stationary density matrix [Formula: see text], for the quantum Markov semigroup [Formula: see text], [Formula: see text], we can define a continuous time stationary quantum Markov process, denoted by [Formula: see text], [Formula: see text] Given an a priori Laplacian operator [Formula: see text], we will present a natural concept of entropy for a class of density matrices on [Formula: see text]. Given a Hermitian operator [Formula: see text] (which plays the role of a Hamiltonian), we will study a version of the variational principle of pressure for [Formula: see text]. A density matrix [Formula: see text] maximizing pressure will be called an equilibrium density matrix. From [Formula: see text] we will derive a new infinitesimal generator [Formula: see text]. Finally, the continuous time quantum Markov process defined by the semigroup [Formula: see text], [Formula: see text], and an initial stationary density matrix, will be called the continuous time equilibrium quantum Markov process for the Hamiltonian [Formula: see text]. It corresponds to the quantum thermodynamical equilibrium for the action of the Hamiltonian [Formula: see text].

Matrix factorizations and pentagon maps

We propose a specific class of matrices that participate in factorization problems that turn out to be equivalent to constant and entwining (non-constant) pentagon, reverse-pentagon or Yang–Baxter maps, expressed in non-commutative variables. In detail, we show that factorizations of order N = 2 matrices of this specific class are equivalent to the homogeneous normalization map . From order N = 3 matrices, we obtain an extension of the homogeneous normalization map, as well as novel entwining pentagon, reverse-pentagon and Yang–Baxter maps.

On the eigenvectors of generalized circulant matrices

In this paper, closed formulas for the eigenvectors of a particular class of matrices generated by generalized permutation matrices, named generalized circulant matrices, are presented.

A New First Finite Class of Classical Orthogonal Polynomials Operational Matrices: An Application for Solving Fractional Differential Equations

In this paper, new operational matrices (OMs) of ordinary and fractional derivatives (FDs) of a first finite class of classical orthogonal polynomials (FFCOP) are introduced. Also, two algorithms are proposed for using the tau and collocation spectral methods (SPMs) to get new approximate solutions to the given fractional differential equations (FDEs). These algorithms convert the given FDEs subject to initial/boundary conditions (I/BCs) into linear or nonlinear systems of algebraic equations that can be solved using appropriate solvers. To demonstrate the robustness, efficiency, and accuracy of the proposed spectral solutions, several illustrative examples are presented. The obtained results show that the proposed algorithms exhibit higher accuracy compared to existing techniques in the literature. Furthermore, an error analysis is provided.

On Rank Selection in Non-Negative Matrix Factorization Using Concordance

The choice of the factorization rank of a matrix is critical, e.g., in dimensionality reduction, filtering, clustering, deconvolution, etc., because selecting a rank that is too high amounts to adjusting the noise, while selecting a rank that is too low results in the oversimplification of the signal. Numerous methods for selecting the factorization rank of a non-negative matrix have been proposed. One of them is the cophenetic correlation coefficient (ccc), widely used in data science to evaluate the number of clusters in a hierarchical clustering. In previous work, it was shown that ccc performs better than other methods for rank selection in non-negative matrix factorization (NMF) when the underlying structure of the matrix consists of orthogonal clusters. In this article, we show that using the ratio of ccc to the approximation error significantly improves the accuracy of the rank selection. We also propose a new criterion, concordance, which, like ccc, benefits from the stochastic nature of NMF; its accuracy is also improved by using its ratio-to-error form. Using real and simulated data, we show that concordance, with a CUSUM-based automatic detection algorithm for its original or ratio-to-error forms, significantly outperforms ccc. It is important to note that the new criterion works for a broader class of matrices, where the underlying clusters are not assumed to be orthogonal.

On the Smith Normal Form of the Least Common Multiple of Matrices from some Class of Matrices

On the Smith Normal Form of the Least Common Multiple of Matrices from some Class of Matrices

Hemoglobin signal network mapping reveals novel indicators for precision medicine

Precision medicine currently relies on a mix of deep phenotyping strategies to guide more individualized healthcare. Despite being widely available and information-rich, physiological time-series measures are often overlooked as a resource to extend insights gained from such measures. Here we have explored resting-state hemoglobin measures applied to intact whole breasts for two subject groups – women with confirmed breast cancer and control subjects – with the goal of achieving a more detailed assessment of the cancer phenotype from a non-invasive measure. Invoked is a novel ordinal partition network method applied to multivariate measures that generates a Markov chain, thereby providing access to quantitative descriptions of short-term dynamics in the form of several classes of adjacency matrices. Exploration of these and their associated co-dependent behaviors unexpectedly reveals features of structured dynamics, some of which are shown to exhibit enzyme-like behaviors and sensitivity to recognized molecular markers of disease. Thus, findings obtained strongly indicate that despite the use of a macroscale sensing method, features more typical of molecular-cellular processes can be identified. Discussed are factors unique to our approach that favor a deeper depiction of tissue phenotypes, its extension to other forms of physiological time-series measures, and its expected utility to advance goals of precision medicine.

Characterization of the unit group of ℤ[G × C2n

Let [Formula: see text] be a group and [Formula: see text] be the cyclic group of order 2. We characterize the unit group [Formula: see text] of the integral group ring [Formula: see text] in terms of the unit group [Formula: see text], for every positive integer [Formula: see text]. As a consequence, for each [Formula: see text], we determine conditions on some class of matrices [Formula: see text]s, to be in GL[Formula: see text]. Using this we calculate the integer solutions of some specific class of symmetric polynomial [Formula: see text] with integer coefficients.

Representatives of similarity classes of matrices over PIDs corresponding to ideal classes

Abstract For a principal ideal domain $A$ , the Latimer–MacDuffee correspondence sets up a bijection between the similarity classes of matrices in $\textrm{M}_{n}(A)$ with irreducible characteristic polynomial $f(x)$ and the ideal classes of the order $A[x]/(f(x))$ . We prove that when $A[x]/(f(x))$ is maximal (i.e. integrally closed, i.e. a Dedekind domain), then every similarity class contains a representative that is, in a sense, close to being a companion matrix. The first step in the proof is to show that any similarity class corresponding to an ideal (not necessarily prime) of degree one contains a representative of the desired form. The second step is a previously unpublished result due to Lenstra that implies that when $A[x]/(f(x))$ is maximal, every ideal class contains an ideal of degree one.

Switching checkerboards in (0,1)-matrices

In order to study M(R,C), the set of binary matrices with fixed row and column sums R and C, we consider submatrices of the form (1001) and (0110), called positive and negative checkerboard respectively. We define an oriented graph of matrices G(R,C) with vertex set M(R,C) and an arc from A to A′ indicates you can reach A′ by switching a negative checkerboard in A to positive. We show that G(R,C) is a directed acyclic graph and identify classes of matrices which constitute unique sinks and sources of G(R,C). Given A,A′∈M(R,C), we give necessary conditions and sufficient conditions on M=A′−A for the existence of a directed path from A to A′.We then consider the special case of M(D), the set of adjacency matrices of graphs with fixed degree distribution D. We define G(D) accordingly by switching negative checkerboards in symmetric pairs. We show that Z2, an approximation of the spectral radius λ1 based on the second Zagreb index, is non-decreasing along arcs of G(D). Also, λ1 reaches its maximum in M(D) at a sink of G(D). We provide simulation results showing that applying successive positive switches to an Erdős-Rényi graph can significantly increase λ1.

Tridiagonal maximum-entropy sampling and tridiagonal masks

The NP-hard maximum-entropy sampling problem (MESP) seeks a maximum (log-) determinant principal submatrix, of a given order, from an input covariance matrix C. We give an efficient dynamic-programming algorithm for MESP when C (or its inverse) is tridiagonal and generalize it to the situation where the support graph of C (or its inverse) is a spider graph with a constant number of legs (and beyond). We give a class of arrowhead covariance matrices C for which a natural greedy algorithm solves MESP. A mask M for MESP is a correlation matrix with which we pre-process C, by taking the Hadamard product M∘C. Upper bounds on MESP with M∘C give upper bounds on MESP with C. Most upper-bounding methods are much faster to apply, when the input matrix is tridiagonal, so we consider tridiagonal masks M (which yield tridiagonal M∘C). We make a detailed analysis of such tridiagonal masks, and develop a combinatorial local-search based upper-bounding method that takes advantage of fast computations on tridiagonal matrices.

Generalized Jordan forms of matrices over division rings

Let [Formula: see text] be a division ring that is algebraic over its center [Formula: see text]. The aim of this paper is to describe certain classes of triangular matrices over [Formula: see text] that are similar to generalized Jordan forms, namely: (1) the class of finite triangular matrices, and (2) the class of infinite triangular matrices whose all diagonal entries, except possibly finitely many, belong to [Formula: see text].