Companion Matrix Research Articles

Factorization and inversion of finite and infinite comrade matrices

Motivated by current applications of the comrade matrices of large order, the infinite invertible comrade matrices are considered. These matrices are a generalization of the companion matrix and a particular case of the bordered tridiagonal matrices. A method based on a simple but suitable factorization is proposed for obtaining, in the finite-dimensional case, a decomposition of the inverse. This procedure can be applied to infinite comrade matrices. Some illustrative examples are given.

Fast QR iterations for unitary plus low rank matrices

Some fast algorithms for computing the eigenvalues of a (block) companion matrix have recently appeared in the literature. In this paper we generalize the approach to encompass unitary plus low rank matrices of the form $$A=U + XY^H$$ where U is a general unitary matrix. Three important cases for applications are U unitary diagonal, U unitary block Hessenberg and U unitary in block CMV form. Our extension exploits the properties of a larger matrix $$\hat{A}$$ obtained by a certain embedding of the Hessenberg reduction of A suitable to maintain its structural properties. We show that $$\hat{A}$$ can be factored as product of lower and upper unitary Hessenberg matrices possibly perturbed in the first k rows, and, moreover, such a data-sparse representation is well suited for the design of fast eigensolvers based on the QR iteration. The resulting eigenvalue algorithm is fast and backward stable.

Exhaustive Search for Various Types of MDS Matrices

MDS matrices are used in the design of diffusion layers in many block ciphers and hash functions due to their optimal branch number. But MDS matrices, in general, have costly implementations. So in search for efficiently implementable MDS matrices, there have been many proposals. In particular, circulant, Hadamard, and recursive MDS matrices from companion matrices have been widely studied. In a recent work, recursive MDS matrices from sparse DSI matrices are studied, which are of interest due to their low fixed cost in hardware implementation. In this paper, we present results on the exhaustive search for (recursive) MDS matrices over GL(4, F2). Specifically, circulant MDS matrices of order 4, 5, 6, 7, 8; Hadamard MDS matrices of order 4, 8; recursive MDS matrices from companion matrices of order 4; recursive MDS matrices from sparse DSI matrices of order 4, 5, 6, 7, 8 are considered. It is to be noted that the exhaustive search is impractical with a naive approach. We first use some linear algebra tools to restrict the search to a smaller domain and then apply some space-time trade-off techniques to get the solutions. From the set of solutions in the restricted domain, one can easily generate all the solutions in the full domain. From the experimental results, we can see the (non) existence of (involutory) MDS matrices for the choices mentioned above. In particular, over GL(4, F2), we provide companion matrices of order 4 that yield involutory MDS matrices, circulant MDS matrices of order 8, and establish the nonexistence of involutory circulant MDS matrices of order 6, 8, circulant MDS matrices of order 7, sparse DSI matrices of order 4 that yield involutory MDS matrices, and sparse DSI matrices of order 5, 6, 7, 8 that yield MDS matrices. To the best of our knowledge, these results were not known before. For the choices mentioned above, if such MDS matrices exist, we provide base sets of MDS matrices, from which all the MDS matrices with the least cost (with respect to d-XOR and s-XOR counts) can be obtained. We also take this opportunity to present some results on the search for sparse DSI matrices over finite fields that yield MDS matrices. We establish that there is no sparse DSI matrix S of order 8 over F28 such that S8 is MDS.

Right core inverses of a product and a companion matrix

In this paper, characterizations of right core inverse are given by one-sided invertibility. The necessary and sufficient conditions, which guarantee that paq has right core inverses, are investigated. Furthermore, characterizations of right core inverses of triangular matrices, matrices and a companion matrix are considered.

Divided Differences Calculus in Matrix Representation

A systematic description of actions of the divided differences operators on power and exponential functions is given. The results of actions of these operators on entire functions are presented by the matrices whose elements are functions of coefficients of a characteristic (pivot) polynomial. Effective algorithms of calculation of the matrices are constructed using the properties of the companion matrix of the pivoting polynomial. Degeneration of the roots of the pivot polynomial reduces the n-order divided differences operator to $$n-1$$ order operator of differentiation. The exponential type invariant functions with respect to higher order derivatives are constructed.

MDS-матрицы, построенные с помощью сопровождающих матриц многочленов и подстановочных матриц

MDS-матрицы, построенные с помощью сопровождающих матриц многочленов и подстановочных матриц

Exponents of primitive companion matrices

A nonnegative matrix $A$ is primitive if for some positive integer $m$ all entries in $A^m$ are positive. The smallest such $m$ is called the exponent of $A$ and written $\exp (A)$. For the class of primitive companion matrices $X$, we find $\exp (A)$ for certain $A \in X$. We find certain values of $m$ for which there is an $n \times n$ primitive companion matrix (for given $n$) with exponent $m$. We also propose open problems for further research.

Frequency-domain approach for the parametric identification of structures with modal interference

This study aims to improve the accuracy of the parametric estimation of systems with modal interference in the frequency domain. The theory of modal identification states that the frequency response function can be expressed as a rational function form by using the curve fitting technique, and the modal parameters can then be estimated from rational fractional coefficients. The conventional common denominator model only indicates the frequency response function of a single-degree-of-freedom system; thus, it cannot acquire the mode shape information. In this paper, we propose the matrix-fractional coefficient model constructed by the frequency response functions of a multiple-degree-of-freedom system for modal identification. To avoid the phenomenon of omitted modes caused by the distortion from modal interference among the vibration modes of systems during modal estimation, we use a system model with higher-order matrix-fractional coefficients, but fictitious modes may be caused by numerical computation. Structural and fictitious modes can be effectively separated by using a different-order constructed stabilization diagram. Modal identification can be implemented by solving the eigenproblem of the companion matrix yielded from least-square estimation. Numerical simulations, including a full model of sedan and one-half railway vehicle in the form of a linear 2D model, as well as the experimental testing of an actual plate, confirm the validity and robustness of the proposed parametric-estimation method for a system with modal interference under noisy conditions.

On the Quaternionic Quadratic Equation $$\varvec{xax+bx+xc+d=0}$$

It is shown that the roots of the quaternion quadratic equation $$xax+bx+xc+d=0$$ can be found by computing the eigenvalues of a companion matrix. This allows to completely discuss the number of solutions and to apply some Gersgorin type theorems. Another consequence is the possibility of constructing equations with a given number of solutions.

Location of Right Eigenvalues of Quaternionic Matrix Polynomials

In this paper, inclusion regions for the right eigenvalues of a quaternionic matrix polynomial are derived from Ostrowski’s type theorem for quaternionic block companion matrices. Furthermore, a right spectral radius inequality and its applications for finding bounds for the right eigenvalues of a quaternionic matrix polynomial is presented. Consequently, these bounds give bounds for the zeros of quaternionic polynomials. Finally, bounds on the eigenvalues of complex matrix polynomials are derived. The comparison between the new bounds and some existing bounds have been illustrated with several examples.

Polynomial equations over octonion algebras

In this paper, we present a complete method for finding the roots of all polynomials of the form [Formula: see text] over a given octonion division algebra. When [Formula: see text] is monic, we also consider the companion matrix and its left and right eigenvalues and study their relations to the roots of [Formula: see text], showing that the right eigenvalues form the conjugacy classes of the roots of [Formula: see text] and the left eigenvalues form a larger set than the roots of [Formula: see text].

Companion matrices and Golden-Fibonacci sequences

We assign a companion sequence to a given companion matrix. Given a companion matrix \(\mathbf{C}\), we make use of the associated companion sequence to provide a simple closed-form expression for \(\mathbf{C}^n\), the nth power of \(\mathbf{C}\). We determine conditions under which a given companion matrix \(\mathbf{C}\) is a primitive matrix. A systematic method for obtaining the limit values of a companion sequence is provided. A new class of primitive companion matrices is introduced for which the limit values of the related companion sequences are connected with the Golden ratio \(\uptau =\frac{1+\sqrt{5}}{2}\). In fact, a new generalization of the well-known \(\mathbf{Q}\)-matrix and the ordinary Fibonacci numbers are presented in this paper. This generalized form of the Fibonacci numbers will be referred to as the Golden-Fibonacci sequence. We show that the limit values of a Golden-Fibonacci sequence are powers of the Golden ratio. We apply primitive companion matrices as encoder matrices and introduce a type of error-correcting codes to be called companion coding. We show that the error-correcting relations of companion coding are connected with the limit values of the related companion sequence.

Quantum Lax Pairs via Dunkl and Cherednik Operators

We establish a direct link between Dunkl operators and quantum Lax matrices $\mathcal L$ for the Calogero--Moser systems associated to an arbitrary Weyl group $W$ (or an arbitrary finite reflection group in the rational case). This interpretation also provides a companion matrix $\mathcal A$ so that $\mathcal L, \mathcal A$ form a quantum Lax pair. Moreover, such an $\mathcal A$ can be associated to any of the higher commuting quantum Hamiltonians of the system, so we obtain a family of quantum Lax pairs. These Lax pairs can be of various sizes, matching the sizes of orbits in the reflection representation of $W$, and in the elliptic case they contain a spectral parameter. This way we reproduce universal classical Lax pairs by D'Hoker-Phong and Bordner-Corrigan-Sasaki, and complement them with quantum Lax pairs in all cases (including the elliptic case, where they were not previously known). The same method, with the Dunkl operators replaced by the Cherednik operators, produces quantum Lax pairs for the generalised Ruijsenaars systems for arbitrary root systems. As one of the main applications, we calculate a Lax matrix for the elliptic $BC_n$ case with nine coupling constants (van Diejen system), thus providing an answer to a long-standing open problem.

Non-sparse Companion Matrices

Given a polynomial $p(z)$, a companion matrix can be thought of as a simple template for placing the coefficients of $p(z)$ in a matrix such that the characteristic polynomial is $p(z)$. The Frobenius companion and the more recently-discovered Fiedler companion matrices are examples. Both the Frobenius and Fiedler companion matrices have the maximum possible number of zero entries, and in that sense are sparse. In this paper, companion matrices are explored that are not sparse. Some constructions of non-sparse companion matrices are provided, and properties that all companion matrices must exhibit are given. For example, it is shown that every companion matrix realization is non-derogatory. Bounds on the minimum number of zeros that must appear in a companion matrix, are also given.

Exchange Rate Pass-through to Prices: VAR Evidence for Albania

This study estimates exchange rate pass-through to prices in Albania using a Vector Autoregressive model from 2000Q1 to 2017 Q1 following Cholesky decomposition. We perform an Augmented Dickey-Fuller test and Phillips-Pherron test to ensure the stationarity of the variables and we estimate the impulse-response functions and the variance decomposition of import, producer, consumer prices and interest rate to oil price /exchange rate shocks. Impulse-response functions indicate an incomplete pass-through of exchange rate to prices and the highest response is of consumer prices and interest rates. Variance decomposition indicates that the variance of import prices is explained by growth rate, its shocks and oil prices shocks. The variance of producer prices is explained by its own shocks, real GDP rate and interest rate whilst consumer prices are explained by its innovations, GDP rate and exchange rate. In order to confirm our results, we order the interest rate before the exchange rate and the findings do not change from the previous results. We perform diagnostic tests for the presence of autocorrelation and the stability of our model and the results show that we fail to reject the null hypothesis for serial autocorrelation and all the roots lie within the companion matrix. However, there is evidence on non-normality in our VAR residuals, but this does not violate our analysis.

Toeplitz nonnegative realization of spectra via companion matrices

Abstract The nonnegative inverse eigenvalue problem (NIEP) is the problem of finding conditions for the existence of an n × n entrywise nonnegative matrix A with prescribed spectrum Λ = {λ 1, . . ., λn}. If the problem has a solution, we say that Λ is realizable and that A is a realizing matrix. In this paper we consider the NIEP for a Toeplitz realizing matrix A, and as far as we know, this is the first work which addresses the Toeplitz nonnegative realization of spectra. We show that nonnegative companion matrices are similar to nonnegative Toeplitz ones. We note that, as a consequence, a realizable list Λ= {λ 1, . . ., λn} of complex numbers in the left-half plane, that is, with Re λi ≤ 0, i = 2, . . ., n, is in particular realizable by a Toeplitz matrix. Moreover, we show how to construct symmetric nonnegative block Toeplitz matrices with prescribed spectrum and we explore the universal realizability of lists, which are realizable by this kind of matrices. We also propose a Matlab Toeplitz routine to compute a Toeplitz solution matrix.

Almost involutory recursive MDS diffusion layers

A recursive MDS matrix is an MDS matrix which can be expressed as a power of some companion matrix. The advantage of such a matrix is that it can be implemented by a single LFSR clocking several times. Such matrices are suitable for the design of diffusion layer in lightweight cryptographic applications. It is known that there do not exist involutory recursive MDS matrices. It means that if a recursive MDS matrix M is considered for the diffusion layer in encryption then the diffusion layer process in both encryption and decryption (if $$M^{-1}$$ needs to be computed) cannot be the same, requiring two different LFSR implementations. In this paper we look at some possibilities of making the implementation of the diffusion layer part in both encryption and decryption to use almost the same circuit (LFSR) by using some special recursive MDS matrices. The difference or the cost of the additional operations/control mechanism used is minimal. In this direction we first discuss two known structures: regular recursive MDS matrices, symmetric recursive MDS matrices. We then propose some other structures called almost involutory recursive MDS matrices which can use the same LFSR for realizing the diffusion layer part in both encryption and decryption. We then present a new method for the direct construction of recursive MDS matrices. Our method gives a new infinite class polynomials that yield recursive MDS matrices. We also present some experimental results and comparison results.

Algebraic linearizations of matrix polynomials

We show how to construct linearizations of matrix polynomials za(z)d0+c0, a(z)b(z), a(z)+b(z) (when deg(b(z))<deg(a(z))), and za(z)d0b(z)+c0 from linearizations of the component parts, a(z) and b(z). This allows the extension to matrix polynomials of a new companion matrix construction.

PC06.03 Tumor Mutation Burden - PRO

PC06.03 Tumor Mutation Burden - PRO

A general framework for boundary equilibrium bifurcations of Filippov systems.

As parameters are varied, a boundary equilibrium bifurcation (BEB) occurs when an equilibrium collides with a discontinuity surface in a piecewise-smooth system of ordinary differential equations. Under certain genericity conditions, at a BEB, the equilibrium either transitions to a pseudo-equilibrium (on the discontinuity surface) or collides and annihilates with a coexisting pseudo-equilibrium. These two scenarios are distinguished by the sign of a certain inner product. Here, it is shown that this sign can be determined from the number of unstable directions associated with the two equilibria by using techniques developed by Feigin. A normal form is proposed for BEBs in systems of any number of dimensions. The normal form involves a companion matrix, as does the leading order sliding dynamics, and so the connection to the stability of the equilibria is explicit. In two dimensions, the parameters of the normal form distinguish, in a simple way, the eight topologically distinct cases for the generic local dynamics at a BEB. A numerical exploration in three dimensions reveals that BEBs can create multiple attractors and chaotic attractors and that the equilibrium at the BEB can be unstable even if both equilibria are stable. The developments presented here stem from seemingly unutilised similarities between BEBs in discontinuous systems (specifically Filippov systems as studied here) and BEBs in continuous systems for which analogous results are, to date, more advanced.