Oscillatory Matrices Research Articles

Spectral theory for bounded banded matrices with positive bidiagonal factorization and mixed multiple orthogonal polynomials

Spectral and factorization properties of oscillatory matrices lead to a spectral Favard theorem for bounded banded matrices, that admit a positive bidiagonal factorization, in terms of sequences of mixed multiple orthogonal polynomials with respect to a set positive Lebesgue–Stieltjes measures. A mixed multiple Gauss quadrature formula with corresponding degrees of precision is given.

Oscillatory banded Hessenberg matrices, multiple orthogonal polynomials and Markov chains

A spectral Favard theorem is derived for bounded banded lower Hessenberg matrices that possess a positive bidiagonal factorization. The rich knowledge concerning the spectral and factorization properties of oscillatory matrices forms the basis of this theorem, which is formulated in terms of sequences of multiple orthogonal polynomials of types I and II, associated with a set of positive Lebesgue-Stieltjes measures. Additionally, a multiple Gauss quadrature is established, and the corresponding degrees of precision are determined. The spectral Favard theorem finds application in the context of Markov chains with transition matrices having (p + 2) diagonals, extending beyond the birth and death scenario, while still maintaining a positive stochastic bidiagonal factorization. In the finite case, the Karlin–McGregor spectral representation is provided, along with the demonstration of recurrent behavior and explicit expressions for the stationary distributions in terms of the orthogonal polynomials. Analogous results are obtained for countably infinite Markov chains. In this case, the Markov chain may not be recurrent, and its characterization is expressed in relation to the first measure. The ergodicity of the Markov chain is explored, taking into consideration the presence of a mass at 1, which corresponds to eigenvalues associated with the right and left eigenvectors.

An improved algorithm for solving an inverse eigenvalue problem for band matrices

The construction of matrices with prescribed eigenvalues is a kind of inverse eigenvalue problems. The authors proposed an algorithm for constructing band oscillatory matrices with prescribed eigenvalues based on the extended discrete hungry Toda equation (Numer. Algor. 75:1079--1101, 2017). In this paper, we develop a new algorithm for constructing band matrices with prescribed eigenvalues based on a generalization of the extended discrete hungry Toda equation. The new algorithm improves the previous algorithm so that the new one can produce more generic band matrices than the previous one in a certain sense. We compare the new algorithm with the previous one by numerical examples. Especially, we show an example of band oscillatory matrices which the new algorithm can produce but the previous one cannot.

Local Law for Singular Values of Oscillatory Matrices

AbstractWe continue the study of spectra of oscillatory random matrices with fully dependent rows. Motivated by the $d$-dimensional skew-shift dynamics from ergodic theory, we introduce the $N\times N$ random matrices $$ \begin{align*} &X_{j,k}=\exp\left(2\pi \textrm{i} \sum_{q=1}^d\ \omega_{j,q} k^q\right), \end{align*}$$where $\{\omega _{j,q}\}_{1\leq j\leq N, 1\leq q\leq d}$ is a collection of i.i.d. random variables and $d$ is a fixed integer. We prove that as $N\to \infty $ the distribution of singular values converges to the local Marchenko–Pastur law up to scales $N^{-\theta _d}$ for an explicit $\theta _d>0$, as long as $d\geq 18$. Our approach provides a novel mechanism—deterministic oscillatory cancellations—for universal spectral laws. The proof blends techniques from random matrix theory, harmonic analysis, and analytic number theory such as strong estimates on the number of solutions to Diophantine equations in the form of Vinogradov’s main conjecture, proved by Bourgain–Demeter–Guth.

Recognition of matrices which are sign-regular of a given order and a generalization of oscillatory matrices

Recognition of matrices which are sign-regular of a given order and a generalization of oscillatory matrices

On the exponent of several classes of oscillatory matrices

Oscillatory matrices were introduced in the seminal work of Gantmacher and Krein. An n×n matrix A is called oscillatory if all its minors are nonnegative and there exists a positive integer k such that all minors of Ak are positive. The smallest k for which this holds is called the exponent of the oscillatory matrix A. Gantmacher and Krein showed that the exponent is always smaller than or equal to n−1. An important and nontrivial problem is to determine the exact value of the exponent. Here we use the successive elementary bidiagonal factorization of oscillatory matrices, and its graph-theoretic representation, to derive an explicit expression for the exponent of several classes of oscillatory matrices, and a nontrivial upper-bound on the exponent for several other classes.

Entrainment to subharmonic trajectories in oscillatory discrete-time systems

A matrix A is called totally positive (TP) if all its minors are positive, and totally nonnegative (TN) if all its minors are nonnegative. A square matrix A is called oscillatory if it is TN and some power of A is TP. A linear time-varying system is called an oscillatory discrete-time system (ODTS) if the matrix defining its evolution at each time k is oscillatory. We analyze the properties of n-dimensional time-varying nonlinear discrete-time systems whose variational system is an ODTS, and show that they have a well-ordered behavior. More precisely, if the nonlinear system is time-varying and T-periodic then any trajectory either leaves any compact set or converges to an ((n−1)T)-periodic trajectory, that is, a subharmonic trajectory. These results hold for any dimension n. The analysis of such systems requires establishing that a line integral of the Jacobian of the nonlinear system is an oscillatory matrix. This is non-trivial, as the sum of two oscillatory matrices is not necessarily oscillatory, and this carries over to integrals. We derive several new sufficient conditions guaranteeing that the line integral of a matrix is oscillatory, and demonstrate how this yields interesting classes of discrete-time nonlinear systems that admit a well-ordered behavior.

Generalized even and odd totally positive matrices

A generalization of the definition of an oscillatory matrix based on the theory of cones is given in this paper. The positivity and simplicity of all the eigenvalues of a generalized oscillatory matrix are proved. Classes of generalized even and odd oscillatory matrices are introduced. Spectral properties of the obtained matrices are studied. Criteria of generalized even and odd oscillation are given. Examples of generalized even and odd oscillatory matrices are presented.

Qualitative Properties of Frequencies and Modes of Vibrating Multi-Span Beam

We begin in this paper by introducing the finite-difference model and the corresponding stiffness matrix for an arbitrarily supported multi-span beam. Next, we derive the condition under which the stiffness matrix becomes sign-oscillatory for a two-span beam. The work is then expanded to cover a three-span beam. In the subsequent discussion, we show that the more generalized case of a multi-span beam can be obtained by applying the method of mathematical induction. Based on the theory of oscillatory matrices, we establish several qualitative properties of natural frequencies and normal modes for the multi-span beam. Finally, a numerical example on a three-span overhanging beam is given to demonstrate the qualitative properties of vibration.

Bidiagonal factorizations and quasi-oscillatory rectangular matrices

A real matrix A, of size m×n, is called totally nonnegative (totally positive) if all its minors are nonnegative (positive). A variant of the Neville elimination process is studied in relation to the existence of a totally nonnegative elementary bidiagonal factorization of A. The class of quasi- oscillatory rectangular matrices, which in the square case contains the oscillatory matrices, is introduced and a characterization of this class of matrices, by incorporating bidiagonal factorization, is showed.

A class of oscillatory matrices with exponent n − 1

A matrix A is called an oscillatory matrix if it is totally nonnegative and there exists a positive integer k such that A k is totally positive. In this paper we describe the class of n × n oscillatory matrices which have maximal exponent equal to n − 1.

Pentadiagonal Oscillatory Matrices with Two Spectrum in Common

We denote the spectrum of an square matrix A by σ(A), and that of the matrix obtained by deleting the first i rows and columns of A by σi(A). It is known that a symmetric pentadiagonal oscillatory (SPO) matrix may be constructed from σ, σ1 and σ2. The pairs σ, σ1 and σ1, σ2 must interlace; the construction is not unique; and the conditions on the data which ensure that A is oscillatory are extremely complicated. Given one SPO matrix A, the paper shows that operations may be applied to A to construct a family of such matrices with σ and σ1 in common. Moreover, given one totally positive (TP) matrix A, we construct a family of TP matrices with σ, σ1 and σ2 in common.

Correlation matrices of yields and total positivity

It has been empirically observed that correlation matrices of forward interest rates have the first three eigenvalues which are simple and their corresponding eigenvectors, termed as shift, slope and curvature respectively, with elements presenting changes of sign in a regular way. These spectral properties are very similar to those exhibited by Strictly Totally Positive and Oscillatory matrices. In the present paper we investigate how these spectral properties are related with those characterizing the correlation matrices considered, i.e. the positivity and the monotonicity of their elements. On the basis of these relations we prove the simplicity of the first two eigenvalues and provide an estimate of the second one.

Two results on basic oscillatory matrices

This paper continues the research of the authors on totally nonnegative and oscillatory matrices. First, the result of the second author stating that the Hadamard product of oscillatory tridiagonal matrices of the same order is again an oscillatory tridiagonal matrix, is extended to the class of basic oscillatory matrices introduced recently. Then it is shown that every oscillatory matrix contains a basic oscillatory matrix as a factor. This explains the role of the class of basic oscillatory matrices within the class of oscillatory matrices.

A remark on oscillatory matrices

In this note we revisit a classical criterion obtained by Gantmacher and Krein for determining when a totally nonnegative matrix is actually oscillatory. A new proof of this criterion is presented by incorporating bidiagonal factorizations of invertible totally nonnegative matrices and utilizing certain associated weighted planar diagrams.

Qualitative Properties of Frequencies and Modes of Beams Modeled by Discrete Systems

ABSTRACTIn this paper, a discrete system model and its equation of motion for beams with arbitrary supports at two ends are established. These supports include elastic, rigid and free supports in translation and rotation directions. Based on theory of oscillatory matrices, a series of qualitative properties of frequencies and modes of this system are derived. The basic properties include: non-zero frequencies are distinct; the ith displacement mode has i - 1 nodes; nodes of ith mode and (i + 1)th mode interlace.Some additional important qualitative properties owned by rotation modes and strain modes are given as well.

Generalized oscillatory matrices

We define a new class of generalized oscillatory matrices, shortly GO-matrices, over a noncommutative ring with identity and a positive part. Similarly to the classical case, this class consists of square generalized totally nonnegative matrices (GTN-matrices) of which some power is generalized totally positive. Using the previously defined ordering of invertible GTN-matrices, we study, in particular, so called basic GO-matrices which form, in a sense, atoms of this ordering.

Interlacing inequalities for totally nonnegative matrices

Suppose λ 1⩾⋯⩾λ n⩾0 are the eigenvalues of an n×n totally nonnegative matrix, and λ ̃ 1⩾⋯⩾ λ ̃ k are the eigenvalues of a k×k principal submatrix. A short proof is given of the interlacing inequalities: λ i⩾ λ ̃ i⩾λ i+n−k, i=1,…,k. It is shown that if k=1,2,n−2,n−1, λ i and λ ̃ j are nonnegative numbers satisfying the above inequalities, then there exists a totally nonnegative matrix with eigenvalues λ i and a submatrix with eigenvalues λ ̃ j . For other values of k, such a result does not hold. Similar results for totally positive and oscillatory matrices are also considered.

A note on the construction of a positive oscillatory matrix with a prescribed spectrum

A necessary and sufficient condition for an n-tuple of real numbers ( λ 1, λ 2, …, λ n ) to be the spectrum of an oscillatory matrix is that λ 1 > λ 2 > … > λ n > 0. Some methods of constructing a positive symmetric oscillatory matrix with spectrum σ = λ 1, λ 2, …, λ n are suggested for theorems concerning oscillatory matrices. In this work, we present a new procedure of constructing such a matrix which is numerically stable and appears to be less expensive.

Totally nonnegative and oscillatory elements in semisimple groups

We generalize the well known characterizations of totally nonnegative and oscillatory matrices, due to F. R. Gantmacher, M. G. Krein, A. Whitney, C. Loewner, M. Gasca, and J. M. Peña to the case of an arbitrary complex semisimple Lie group.