Rational Matrix Research Articles

RATIONAL REPRESENTATIONS AND RATIONAL GROUP ALGEBRA OF VZ $\boldsymbol {p}$ -GROUPS

Abstract In this article, we study rational matrix representations of VZ p-groups (p is any prime). Using our findings on VZ p-groups, we explicitly obtain all inequivalent irreducible rational matrix representations of all p-groups of order $\leq p^4$ . Furthermore, we establish combinatorial formulae to determine the Wedderburn decompositions of rational group algebras for VZ p-groups and all p-groups of order $\leq p^4$ , ensuring simplicity in the process.

On a case of -linear conjugation problem on the unit circle

The R -linear conjugation problem on the unit circle with rational coefficients is solved. The problem is equivalently reduced to the vector–matrix C -linear conjugation problem for which the canonical matrix is effectively constructed. The applied method is based on the two-fold application of G.N.Chebotarev's approach to the factorization of rational matrix functions.

Numerical semigroups from rational matrices I: power-integral matrices and nilpotent representations

Our aim in this paper is to initiate the study of exponent semigroups for rational matrices. We prove that every numerical semigroup is the exponent semigroup of some rational matrix. We also obtain lower bounds on the size of such matrices and discuss the related class of power-integral matrices.

Wiener-Hopf Indices of Unitary-Valued Functions on the Imaginary Axis

This paper is concerned with the Wiener-Hopf indices of unitary-valued rational matrix functions on the imaginary axis. These indices play a role in the Fredholm theory for Wiener-Hopf integral operators. Our main result gives formulas for the Wiener-Hopf indices in terms of the matrices appearing in realizations of the factors in a Douglas-Shapiro-Shields factorization of the unitary-valued function. Two approaches to this problem are presented: one direct approach using operator theoretic methods, and a second approach using the Cayley transform which allows to use results for an analogous problem regarding unitary-valued functions on the unit circle and corresponding Toeplitz operators.

Counting vanishing matrix-vector products

Consider the following parameterized counting variation of the classic subset sum problem, which arises notably in the context of higher homotopy groups of topological spaces. Let v∈Qd be a rational vector, (T1,T2…,Tm) a list of d×d rational matrices, S∈Qh×d a rational matrix not necessarily square and k a parameter. The goal is to compute the number of ways one can choose k matrices Ti1,Ti2,…,Tik from the list such that STik⋯Ti1v=0∈Qh.In this paper, we show that this problem is #W[2]-hard for parameter k. As a consequence, computing the k-th homotopy group of a d-dimensional 1-connected topological space for d>3 is #W[2]-hard for parameter k. We also discuss a decision version of the problem and its several modifications for which we show W[1]/W[2]-hardness. This is in contrast to the parameterized k-sum problem, which is only W[1]-hard (Abboud-Lewi-Williams, ESA'14). In addition, we show that the decision version of the problem without parameter is an undecidable problem, and we give a fixed-parameter tractable algorithm for matrices of bounded size over finite fields, parameterized by the matrix dimensions and the order of the field.

A Toeplitz-like operator with rational matrix symbol having poles on the unit circle: Fredholm characteristics

A Toeplitz-like operator with rational matrix symbol having poles on the unit circle: Fredholm characteristics

Unsteady Lifting-Line Theory for Camber Morphing Wings State-Space Aeroelastic Modeling

A novel frequency-domain analytical-numerical model for the aerodynamic solution of camber morphing wings is developed. The model combines an unsteady lifting-line formulation with Küssner–Schwarz aerodynamic theory to provide the pressure distribution and thus the transcendental aerodynamic matrix that relates the generalized aerodynamic forces to the Lagrangian coordinates of a given wing structural dynamics model. The state-space form of the aerodynamic loads is obtained from the rational matrix approximation of the aerodynamic matrix. This, combined with the wing structural dynamics operator, can readily provide the state-space form of the aeroelastic problem. To assess the accuracy of the proposed aerodynamic solution method, numerical investigations are performed. These consist of its application to several conventional wing configurations and wings with camber morphing, followed by the comparisons of the corresponding solutions with the predictions given by a well-validated panel-method solver for potential flows. These results are shown to be in very good agreement, thus demonstrating that the proposed approach can capture the effects due to wing tapering, sweep angle, and camber morphing, while requiring a remarkably lower computational effort. The excellent accuracy of a few-pole finite-state approximation of the aerodynamic matrix is finally proven for a swept wing with camber morphing.

On the Exact Spectral Factorization of Rational Matrix Functions with Applications to Paraunitary Filter Banks

On the Exact Spectral Factorization of Rational Matrix Functions with Applications to Paraunitary Filter Banks

A vector decomposition approach to F–M model realization and its application to robust observed-state feedback control

This work presents a novel approach based on vector decomposition for realizing the Fornasini–Marchesini (F–M) model in multidimensional (n-D) systems. The primary focus is on resolving coefficient dependencies within the n-D rational transfer function matrix (TFM) of a multiple-input multiple-output (MIMO) system, aiming to generate a lower-order F–M model. The proposed method is further extended to robust observational state feedback control. To achieve this, we introduce innovative sufficient conditions for designing a low-dimensional resolvent invariant space (RIS) and outline the corresponding low-order F–M model realization process. By incorporating the polynomial vector dependency technique, we comprehensively address coefficient dependencies, leading to the construction of an even lower-order F–M model and contributing to the simplification of analytical complexity. To accommodate generic rational TFMs, we propose a matrix fractional description (MFD)-based process. Finally, we apply the developed F–M model implementation method to derive descriptor multiple affine representations (DMARs). This application proves instrumental in the robust observed-state feedback control design for rationally uncertain systems, enhancing the overall effectiveness of the robust observed-state feedback control design.

Computing a compact local Smith–McMillan form

We define a compact local Smith–McMillan form of a rational matrix R ( λ ) as the diagonal matrix whose diagonal elements are the nonzero entries of a local Smith-McMillan form of R ( λ ) . We show that a recursive rank search procedure, applied to a block-Toeplitz matrix built on the Laurent expansion of R ( λ ) around an arbitrary complex point λ 0 , allows us to compute a compact local Smith-McMillan form of that rational matrix R ( λ ) at the point λ 0 , provided we keep track of the transformation matrices used in the rank search. It also allows us to recover the root polynomials of a polynomial matrix and root vectors of a rational matrix, at an expansion point λ 0 . Numerical tests illustrate the promising performance of the resulting algorithm.

Deciding whether a lattice has an orthonormal basis is in co-NP

AbstractWe show that the problem of deciding whether a given Euclidean lattice L has an orthonormal basis is in NP and co-NP. Since this is equivalent to saying that L is isomorphic to the standard integer lattice, this problem is a special form of the lattice isomorphism problem, which is known to be in the complexity class SZK. We achieve this by deploying a result on characteristic vectors by Elkies that gained attention in the context of 4-manifolds and Seiberg-Witten equations, but seems rather unnoticed in the algorithmic lattice community. On the way, we also show that for a given Gram matrix $$G \in \mathbb {Q}^{n \times n}$$ G ∈ Q n × n , we can efficiently find a rational lattice that is embedded in at most four times the initial dimension n, i.e. a rational matrix $$B \in \mathbb {Q}^{4n \times n}$$ B ∈ Q 4 n × n such that $$B^\intercal B = G$$ B ⊺ B = G .

Structured eigenvalue backward errors for rational matrix functions with symmetry structures

Structured eigenvalue backward errors for rational matrix functions with symmetry structures

Fiedler Linearizations of Rectangular Rational Matrix Functions

Fiedler Linearizations of Rectangular Rational Matrix Functions

A Toeplitz-like operator with rational matrix symbol having poles on the unit circle: Invertibility and Riccati equations

This paper is a continuation of the work on unbounded Toeplitz-like operators TΩ with rational matrix symbol Ω initiated in Groenewald et al. (2021) [11], where a Wiener-Hopf type factorization of Ω is obtained and used to determine when TΩ is Fredholm and to compute the Fredholm index in case TΩ is Fredholm. Due to the high level of non-uniqueness and complicated form of the Wiener-Hopf type factorization, it does not appear useful in determining when TΩ is invertible. In the present paper we use state space methods to characterize invertibility of TΩ in terms of the existence of a stabilizing solution of an associated nonsymmetric discrete algebraic Riccati equation, which in turn leads to a pseudo-canonical factorization of Ω and concrete formulas of TΩ−1.

State-space model realization for non-commensurate fractional-order systems based on Gleason’s problem

This paper presents a novel state-space model (SSM) realization approach for non-commensurate fractional-order (NCFO) systems. A new notion of the resolvent invariant backward shift space of Gleason’s problem associated with the NCFO rational transfer function matrix is introduced firstly, based on which a necessary and sufficient condition for the existence of the NCFO-SSM realization is given. Then, a sufficient condition for the construction of the resolvent invariant backward shift space of NCFO Gleason’s problem is presented, and the corresponding realization algorithms are developed. In order to obtain a lower-dimensional NCFO-SSM, further improvement based on a polynomial description is studied. Finally, symbolic and numerical examples are provided to illustrate the details as well as the effectiveness of the proposed NCFO-SSM realization approach.

Spectral number of 3-Bernoulli convolutions on and Sierpinski-type measures on

Let be a class of Sierpinski-type measures generated by a pair , where with and and . And let µ b be the 3-Bernoulli convolutions on determined by the pair with . It has been shown that and µ b admit an infinitely many exponential mutually orthogonal system if and only if with , and with respectively. In this paper, we will study the maximal number of exponentials of orthogonal sets of and which we call the spectral number of or µ b . In view of the connection of orthogonality between Sierpinski-type measures on and 3-Bernoulli convolutions µ b on , we study the spectral number of µ b according to the cut-off point . Based on the results for µ b , we give a classification on the spectral number of all Sierpinski-type measures except for the case that at least one and it is not in the form of with . In addition, we provide a structure theorem on the exponential orthogonal sets in for , and at least one and that in for and . To the end, we give an explicit representation on the maximal orthogonal set of exponentials for a class of Moran measures µ w by defining a mixed tree map over a symbol space. As an application, all maximal orthogonal sets of exponentials of with the rational can be explicitly expressed. This result improves the characterization of maximal orthogonal set of exponentials for the integral matrix to that for the rational matrix .

Low-Memory Krylov Subspace Methods for Optimal Rational Matrix Function Approximation

We describe a Lanczos-based algorithm for approximating the product of a rational matrix function with a vector. This algorithm, which we call the Lanczos method for optimal rational matrix function approximation (Lanczos-OR), returns the optimal approximation from a given Krylov subspace in a norm depending on the rational function's denominator, and can be computed using the information from a slightly larger Krylov subspace. We also provide a low-memory implementation which only requires storing a number of vectors proportional to the denominator degree of the rational function. Finally, we show that Lanczos-OR can be used to derive algorithms for computing other matrix functions, including the matrix sign function and quadrature based rational function approximations. In many cases, it improves on the approximation quality of prior approaches, including the standard Lanczos method, with little additional computational overhead.

Smith Normal Form and the generalized spectral characterization of oriented graphs

Spectral characterization of graphs is an important topic in spectral graph theory. An oriented graph Gσ is obtained from a simple undirected graph G by assigning to every edge of G a direction so that Gσ becomes a directed graph. The skew-adjacency matrix of an oriented graph Gσ is a real skew-symmetric matrix S(Gσ)=(sij), where sij=−sji=1 if (i,j) is an arc; sij=sji=0 otherwise. Let Gσ and Hτ be two oriented graphs whose skew-adjacency matrices are S(Gσ) and S(Hτ), respectively. We say Gσ is R-cospectral to Hτ if tJ−S(Gσ) and tJ−S(Hτ) have the same spectrum for any t∈R, where J is the all-ones matrix. An oriented graph Gσ is said to be determined by the generalized skew spectrum (DGSS for short), if any oriented graph which is R-cospectral to Gσ is isomorphic to Gσ. Let W(Gσ)=[e,S(Gσ)e,S2(Gσ)e,…,Sn−1(Gσ)e] be the skew-walk-matrix of Gσ, where e is the all-ones vector. A theorem of Qiu, Wang and Wang [9] states that if Gσ is a self-converse oriented graph and 2−⌊n2⌋det⁡W(Gσ) is odd and square-free, then Gσ is DGSS. In this paper, based on the Smith Normal Form of the skew-walk-matrix of Gσ we obtain our main result: Let q be a prime and Gσ be a self-converse oriented graph on n vertices with det⁡W(Gσ)≠0. Assume that rankq(W(Gσ))=n−1 if q is an odd prime, and rankq(W(Gσ))=⌈n2⌉ if q=2. If Q is a regular rational orthogonal matrix satisfying QTS(Gσ)Q=S(Hτ) for some oriented graph Hτ, then the level of Q divides dn(W(Gσ))q, where dn(W(Gσ)) is the n-th invariant factor of W(Gσ). Consequently, it leads to an easier way to prove Qiu, Wang and Wang's theorem above.

Graphs with at most one Generalized Cospectral Mate

Let $G$ be an $n$-vertex graph with adjacency matrix $A$, and $W=[e,Ae,\ldots,A^{n-1}e]$ be the walk matrix of $G$, where $e$ is the all-one vector. In Wang [J. Combin. Theory, Ser. B, 122 (2017): 438-451], the author showed that any graph $G$ is uniquely determined by its generalized spectrum (DGS) whenever $2^{-\lfloor n/2 \rfloor}\det W$ is odd and square-free. In this paper, we introduce a large family of graphs$$\mathcal{F}_n=\{\text{$n$-vertex graphs~} G\colon\, 2^{-\lfloor n/2 \rfloor}\det W =p^2b\text{ and rank}W=n-1\text{ over } \mathbb{Z}/p\mathbb{Z}\},$$where $b$ is odd and square-free, $p$ is an odd prime and $p\nmid b$. We prove that any graph in $\mathcal{F}_n$ either is DGS or has exactly one generalized cospectral mate up to isomorphism. Moreover, we show that the problem of finding the generalized cospectral mate for a graph in $\mathcal{F}_n$ is equivalent to that of generating an appropriate rational orthogonal matrix from a given integral vector. This equivalence essentially depends on a surprising property of graphs in terms of generalized spectra, which states that any symmetric integral matrix generalized cospectral with the adjacency matrix of some graph must be an adjacency matrix. Based on this equivalence, we develop an efficient algorithm to decide whether a given graph in $\mathcal{F}_n$ is DGS and further to find the unique generalized cospectral mate when it is not. We give some experimental results on graphs with at most 20 vertices, which suggest that $\mathcal{F}_n$ may have a positive density (nearly $3\%$) and possibly almost all graphs in $\mathcal{F}_n$ are DGS as $n\rightarrow \infty$. This gives supporting evidence for Haemers' conjecture that almost all graphs are determined by their spectra.

Structural backward stability in rational eigenvalue problems solved via block Kronecker linearizations

In this paper we study the backward stability of running a backward stable eigenstructure solver on a pencil $$S(\lambda )$$ that is a strong linearization of a rational matrix $$R(\lambda )$$ expressed in the form $$R(\lambda )=D(\lambda )+ C(\lambda I_\ell -A)^{-1}B$$ , where $$D(\lambda )$$ is a polynomial matrix and $$C(\lambda I_\ell -A)^{-1}B$$ is a minimal state-space realization. We consider the family of block Kronecker linearizations of $$R(\lambda )$$ , which have the following structure $$\begin{aligned} S(\lambda ):=\left[ \begin{array}{ccc} M(\lambda ) &{} {\widehat{K}}_2^T C &{} K_2^T(\lambda ) \\ B {\widehat{K}}_1 &{} A- \lambda I_\ell &{} 0\\ K_1(\lambda ) &{} 0 &{} 0 \end{array}\right] , \end{aligned}$$ where the blocks have some specific structures. Backward stable eigenstructure solvers, such as the QZ or the staircase algorithms, applied to $$S(\lambda )$$ will compute the exact eigenstructure of a perturbed pencil $$\widehat{S}(\lambda ):=S(\lambda )+\varDelta _S(\lambda )$$ and the special structure of $$S(\lambda )$$ will be lost, including the zero blocks below the anti-diagonal. In order to link this perturbed pencil with a nearby rational matrix, we construct in this paper a strictly equivalent pencil $$\widetilde{S}(\lambda )=(I-X)\widehat{S}(\lambda )(I-Y)$$ that restores the original structure, and hence is a block Kronecker linearization of a perturbed rational matrix $${{\widetilde{R}}}(\lambda ) = {{\widetilde{D}}}(\lambda )+ {{\widetilde{C}}}(\lambda I_\ell - {{\widetilde{A}}})^{-1} {{\widetilde{B}}}$$ , where $${{\widetilde{D}}}(\lambda )$$ is a polynomial matrix with the same degree as $$D(\lambda )$$ . Moreover, we bound appropriate norms of $${{\widetilde{D}}}(\lambda )- D(\lambda )$$ , $${{\widetilde{C}}} - C$$ , $${{\widetilde{A}}} - A$$ and $${{\widetilde{B}}} - B$$ in terms of an appropriate norm of $$\varDelta _S(\lambda )$$ . These bounds may be, in general, inadmissibly large, but we also introduce a scaling that allows us to make them satisfactorily tiny, by making the matrices appearing in both $$S(\lambda )$$ and $$R(\lambda )$$ have norms bounded by 1. Thus, for this scaled representation, we prove that the staircase and the QZ algorithms compute the exact eigenstructure of a rational matrix $${{\widetilde{R}}}(\lambda )$$ that can be expressed in exactly the same form as $$R(\lambda )$$ with the parameters defining the representation very near to those of $$R(\lambda )$$ . This shows that this approach is backward stable in a structured sense. Several numerical experiments confirm the obtained backward stability results.