Polynomial Matrix Research Articles

On the eigenvalues and Seidel eigenvalues of chain graphs

In this paper, we primarily focus on the eigenvalues of the adjacency matrix and Seidel matrix of chain graphs, referred to as eigenvalues and Seidel eigenvalues of these graphs, respectively. Firstly, we utilize the characteristic polynomial of the adjacency matrix of a chain graph to construct infinite pairs of non-isomorphic cospectral chain graphs. Next, we determine the inertia of the Seidel matrix of a chain graph and establish an interval that does not contain the Seidel eigenvalues of chain graphs. Lastly, we characterize chain graphs with distinct Seidel eigenvalues.

Generalized Inversion of Nonlinear Operators

Inversion of operators is a fundamental concept in data processing. Inversion of linear operators is well studied, supported by established theory. When an inverse either does not exist or is not unique, generalized inverses are used. Most notable is the Moore–Penrose inverse, widely used in physics, statistics, and various fields of engineering. This work investigates generalized inversion of nonlinear operators. We first address broadly the desired properties of generalized inverses, guided by the Moore–Penrose axioms. We define the notion for general sets and then a refinement, termed pseudo-inverse, for normed spaces. We present conditions for existence and uniqueness of a pseudo-inverse and establish theoretical results investigating its properties, such as continuity, its value for operator compositions and projection operators, and others. Analytic expressions are given for the pseudo-inverse of some well-known, non-invertible, nonlinear operators, such as hard- or soft-thresholding and ReLU. We analyze a neural layer and discuss relations to wavelet thresholding. Next, the Drazin inverse, and a relaxation, are investigated for operators with equal domain and range. We present scenarios where inversion is expressible as a linear combination of forward applications of the operator. Such scenarios arise for classes of nonlinear operators with vanishing polynomials, similar to the minimal or characteristic polynomials for matrices. Inversion using forward applications may facilitate the development of new efficient algorithms for approximating generalized inversion of complex nonlinear operators.

Moments of zeta and correlations of divisor‐sums: Stratification and Vandermonde integrals

AbstractWe refine a recent heuristic developed by Keating and the second author. Our improvement leads to a new integral expression for the conjectured asymptotic formula for shifted moments of the Riemann zeta‐function. This expression is analogous to a formula, recently discovered by Brad Rodgers and Kannan Soundararajan, for moments of characteristic polynomials of random matrices from the unitary group.

New Results on Equivalence of Multivariate Polynomial Matrices

New Results on Equivalence of Multivariate Polynomial Matrices

Lattice factorization based causal symmetric paraunitary matrix extension and construction of symmetric orthogonal multiwavelets

In this paper, we propose a lattice factorization based symmetric paraunitary matrix extension method to design a causal symmetric paraunitary multifilter banks(PUMFBs) and construct compactly supported symmetric orthogonal multiwavelets by using the method.First, we transform a Laurent polynomial matrix consisting of polyphase components of a given symmetric orthogonal multifilter into a Laurent polynomial matrix with a simple structure, and then obtain a lattice factorization based symmetric paraunitary extension matrix by factorizing the transformed Laurent polynomial matrix and extending the constant matrix of the factorization. We also construct symmetric orthogonal multiwavelets by the proposed symmetric paraunitary matrix extension method. Finally, some examples are given to illustrate the proposed symmetric paraunitary matrix extension method.

Bivariate Polynomial Matrix and Smith Form

Matrix equivalence plays a pivotal role in multidimensional systems, which are typically represented by multivariate polynomial matrices. The Smith form of matrices is one of the important research topics in polynomial matrices. This article mainly investigates the Smith forms of several types of bivariate polynomial matrices and has successfully derived several necessary and sufficient conditions for matrix equivalence.

Linearizations of Cubic Two-Parameter Eigenvalue Problems and Its Vector Space

The paper considers the study of linearization techniques of cubic two-parameter matrix polynomial (CTMP). We analyze vector spaces of linearization of CTMP, namely ansatz vector space and double ansatz vector space. We also consider cubic two-parameter eigenvalue problem (CTEP) to study their linearization classes. A unified framework on linearization of CTMP will be established. The conditions under which a matrix pencil in the ansatz spaces is a linearization of CTMP will also be derived. Moreover, using these linearization techniques, CTEP is first converted into a singular linear two-parameter eigenvalue problem (2PEP) of larger size so that existing numerical method for (2PEP) can be applied.

On Matrix Multiplication and Polynomial Identity Testing

On Matrix Multiplication and Polynomial Identity Testing

Matrix product and quasi-twisted codes in one class

Many classical constructions, such as Plotkin’s and Turyn’s, were generalized by matrix product (MP) codes. Quasi-twisted (QT) codes, on the other hand, form an algebraically rich structure class that contains many codes with best-known parameters. We significantly extend the definition of MP codes to establish a broader class of generalized matrix product (GMP) codes that contains QT codes as well. We propose a generator matrix formula for any linear GMP code and provide a condition for determining the code size. We prove that any QT code has a GMP structure. Then we show how to build a generator polynomial matrix for a QT code from its GMP structure, and vice versa. Even though the class of QT codes contains many codes with best-known parameters, we present different examples of GMP codes with best-known parameters that are neither MP nor QT. Two different lower bounds on the minimum distance of GMP codes are presented; they generalize their counterparts in the MP codes literature. The second proposed lower bound replaces the non-singular by columns matrix with a less restrictive condition. Some examples are provided for comparing the two proposed bounds, as well as showing that these bounds are tight.

LeCaSiM: Learning Causal Structure via Inverse of M-Matrices with Adjustable Coefficients

The objective of causal discovery is to uncover the causal relationships among natural phenomena or human behaviors, thus establishing the basis for subsequent prediction and inference. Traditional ways to reveal the causal structure between variables, such as interventions or random trials, are often deemed costly or impractical. With the diversity of means of data acquisition and abundant data, approaches that directly learn causal structure from observational data are attracting more interest in academia. Unfortunately, learning a directed acyclic graph (DAG) from samples of multiple variables has been proven to be a challenging problem. Recent proposals leverage the continuous optimization method to discover the structure of the target DAG from observational data using an acyclic constraint function. Although it has an elegant mathematical form, its optimization process is prone to numerical explosion or the disappearance of high-order gradients of constraint conditions, making the convergence conditions unstable and the result deviating from the true structure. To tackle these issues, we first combine the existing work to give a series of algebraic equivalent conditions for a DAG and its corresponding brief proofs, then leverage the properties of the M-matrix to derive a new acyclic constraint function. Based on the utilization of this function and linear structural equation models, we propose a novel causal discovery algorithm called LeCaSiM, which employs continuous optimization. Our algorithm exploits the spectral radius of the adjacency matrix to dynamically adjust the coefficients of the matrix polynomial, effectively solving the above problems. We conduct extensive experiments on both synthetic and real-world datasets, the results show our algorithm outperforms the existing algorithms on both computation time and accuracy under the same settings.

Independence and orthogonality of algebraic eigenvectors over the max-plus algebra

The max-plus algebra R ∪ { − ∞ } is a semiring with the two operations: addition a ⊕ b := max ( a , b ) and multiplication a ⊗ b := a + b . Roots of the characteristic polynomial of a max-plus matrix are called algebraic eigenvalues. Recently, algebraic eigenvectors with respect to algebraic eigenvalues were introduced as a generalized concept of eigenvectors. In this paper, we present properties of algebraic eigenvectors analogous to those of eigenvectors in the conventional linear algebra. First, we prove that for generic matrices algebraic eigenvectors with respect to distinct algebraic eigenvalues are linearly independent. We further prove that algebraic eigenvectors of symmetric matrices with respect to distinct algebraic eigenvalues are orthogonal to each other.

Row or Column Completion of Polynomial Matrices of Given Degree

Row or Column Completion of Polynomial Matrices of Given Degree

Three New Proofs of the Theorem rank f(M) + rank g(M) = rank (f,g)(M) + rank [f,g](M)

It is well known that in C[X], the product of two polynomials is equal to the product of their greatest common divisor and their least common multiple. In a recent paper, we proved a similar relation between the ranks of matrix polynomials. More precisely, the sum of the ranks of two matrix polynomials is equal to the sum of the rank of the greatest common divisor of the polynomials applied to the respective matrix and the rank of the least common multiple of the polynomials applied to the respective matrix. In this paper, we present three new proofs for this result. In addition to these, we present two more applications.

Parametrizing spectral filters for nonsingular polynomial matrices

Two polynomial matrices are spectrally equivalent if they have the same finite and infinite elementary divisors. Spectral equivalence can be characterized using very special polynomial matrices called spectral filters. The main results in this paper concern the parametrization of spectral filters for any given pair of nonsingular spectrally equivalent polynomial matrices of arbitrary degree d. First we parametrize the set of their spectral filters of degree less than d when the given spectrally equivalent matrices have nonsingular leading coefficients. The parameter space is the subset of invertible matrices of the centralizer of any of their linearizations. When the leading coefficients are singular, we provide a parametrization of the set of spectral filters of degree d−1 and show how to obtain the spectral filters of degree less than d−1. Now the parameter space is the subset of invertible matrices of the centralizer of any linearization of the reversals with respect to a scalar that is not an eigenvalue of the given matrices. Ideas from the Theory of Linear Control Systems are used to introduce left and right realizations of matrix polynomials. These are the building blocks on which the parameterization of spectral filters is constructed. The results are illustrated with examples.

Fully actuated system approach for high-order linear systems with high-order input derivatives

This paper presents a frequency-domain method for transforming a class of high-order linear systems with high-order derivatives of input into the corresponding full-actuated system (FAS) models. Based on the coprime factorisation theory, it is proved that under the condition that the factorisation polynomial matrix is column reduced, a direct method can be given to transform the original system into an FAS model. Regard the non-square problem faced by the direct method, two different methods are given to present the explicit expression of the transformed state. The first one is given for the case the order of system is higher than that of input, in which the system is transformed into the corresponding observability canonical form. The other one considers the opposite case, in which the model reduction method is adopted. Three numerical examples are given to illustrate the proposed theories.

Using easy coefficients conjecture for rotation symmetric Boolean functions

A Boolean function in n variables is rotation symmetric (RS) if it is invariant under powers of ρ(x1,…,xn)=(x2,…,xn,x1). An RS function is called monomial rotation symmetric (MRS) if it is generated by applying powers of ρ to a single monomial. Completing earlier research on special cases, the author showed in 2018 that for any RS function fn in n variables, the sequence of Hamming weights wt(fn) for all values of n satisfies a linear recurrence relation. It was also proved that the associated recursion polynomial could be explicitly calculated as the minimal polynomial of a rules matrix and an algorithm for computing the rules matrix was explained. Examples showed that the usual formula which gives the values of wt(fn)−2n−1 as a linear combination of powers of the irrational roots of the minimal polynomial has simple coefficients which are all 1/2 if the multiset made up of the roots of the characteristic polynomial of the rules matrix is used instead, no matter what the degree of the Boolean function is. The conjecture that this is always true is called the Easy Coefficients Conjecture (ECC). The special case of MRS quadratics was proved in an earlier paper. The present paper gives some ECC applications valid for any function for which the ECC is true, even though there is a proof only for the quadratic MRS case so far.

Frequency-Domain Identification of Discrete-Time Systems using Sum-of-Rational Optimization

This paper proposes a new computationally tractable method to fit coefficients of a fixed-order discrete-time transfer function to a measured frequency response, with stability guaranteed. The problem is formulated as a non-convex global sum-of-rational optimization problem whose objective function is the sum of weighted squared residuals at each observed frequency datapoint. Stability is enforced using a polynomial matrix inequality constraint. The problem is solved by a moment-sum-of-squares hierarchy of semidefinite programs through a framework for sum-of-rational-functions optimization. Convergence of the moment-sum-of-squares program is guaranteed as the bound on the degree of the sum-of-squares polynomials approaches infinity. The performance of the proposed method is demonstrated using numerical simulation examples.

Does Mccutcheon’s Mortality Polynomial Matrix Actually Account for Mortality Decline at Ten Years?

Does Mccutcheon’s Mortality Polynomial Matrix Actually Account for Mortality Decline at Ten Years?

On the logarithmic derivative of characteristic polynomials for random unitary matrices

AbstractLet be a random unitary matrix of size , distributed with respect to the Haar measure on . Let be the characteristic polynomial of . We prove that for close to the unit circle, can be approximated using zeros of very close to , with a typically controllable error term. This is an analogue of a result of Selberg for the Riemann zeta‐function. We also prove a mesoscopic central limit theorem for away from the unit circle, and this is an analogue of a result of Lester for zeta.

Exact Correlations in Topological Quantum Chains

Although free-fermion systems are considered exactly solvable, they generically do not admit closed expressions for nonlocal quantities such as topological string correlations or entanglement measures. We derive closed expressions for such quantities for a dense subclass of certain classes of topological fermionic wires (classes BDI and AIII). Our results also apply to spin chains called generalised cluster models. While there is a bijection between general models in these classes and Laurent polynomials, restricting to polynomials with degenerate zeros leads to a plethora of exact results: (1) we derive closed expressions for the string correlation functions - the order parameters for the topological phases in these classes; (2) we obtain an exact formula for the characteristic polynomial of the correlation matrix, giving insight into ground state entanglement; (3) the latter implies that the ground state can be described by a matrix product state (MPS) with a finite bond dimension in the thermodynamic limit - an independent and explicit construction for the BDI class is given in a concurrent work [Phys. Rev. Res. 3 (2021), 033265, 26 pages, arXiv:2105.12143]; (4) for BDI models with even integer topological invariant, all non-zero eigenvalues of the transfer matrix are identified as products of zeros and inverse zeros of the aforementioned polynomial. General models in these classes can be obtained by taking limits of the models we analyse, giving a further application of our results. To the best of our knowledge, these results constitute the first application of Day's formula and Gorodetsky's formula for Toeplitz determinants to many-body quantum physics.