scholarly journals Nonsparse companion Hessenberg matrices

2021 ◽  
Vol 37 ◽  
pp. 193-210
Author(s):  
Alberto Borobia ◽  
Roberto Canogar
Keyword(s):  
Linear Algebra ◽  
Characteristic Polynomial ◽  
Systematic Study ◽  
Polynomial Basis ◽  
Companion Matrix ◽  
Simple Type ◽  
Hessenberg Matrix ◽  
Monomial Basis ◽  
Companion Matrices ◽  
Chebyshev Basis

In recent years, there has been a growing interest in companion matrices. Sparse companion matrices are well known: every sparse companion matrix is equivalent to a Hessenberg matrix of a particular simple type. Recently, Deaett et al. [Electron. J. Linear Algebra, 35:223--247, 2019] started the systematic study of nonsparse companion matrices. They proved that every nonsparse companion matrix is nonderogatory, although not necessarily equivalent to a Hessenberg matrix. In this paper, the nonsparse companion matrices which are unit Hessenberg are described. In a companion matrix, the variables are the coordinates of the characteristic polynomial with respect to the monomial basis. A PB-companion matrix is a generalization, in the sense that the variables are the coordinates of the characteristic polynomial with respect to a general polynomial basis. The literature provides examples with Newton basis, Chebyshev basis, and other general orthogonal bases. Here, the PB-companion matrices which are unit Hessenberg are also described.

scholarly journals Linear fractional transforms of companion matrices

1979 ◽  
Vol 20 (2) ◽  
pp. 129-132 ◽  
Author(s):  
N. J. Young
Keyword(s):  
Characteristic Polynomial ◽  
Canonical Form ◽  
Stability Theory ◽  
Companion Matrix ◽  
Arbitrary Field ◽  
Companion Matrices ◽  
Image Position ◽  
Fractional Transforms

Questions about polynomials can be turned into questions about matrices by associating with the polynomial(over an arbitrary field) its companion matrixwhich has p/an as its characteristic polynomial. This technique is often used in stability theory, as indicated in [1]; companion matrices also occur in the theory of the rational canonical form.

scholarly journals Non-sparse Companion Matrices

2019 ◽  
Vol 35 ◽  
pp. 223-247 ◽  
Author(s):  
Louis Deaett ◽  
Jonathan Fischer ◽  
Colin Garnett ◽  
Kevin Vander Meulen
Keyword(s):  
Characteristic Polynomial ◽  
Companion Matrix ◽  
Number Of Zeros ◽  
Minimum Number ◽  
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.

XXIII.—Genetic Algebras

1940 ◽  
Vol 59 ◽  
pp. 242-258 ◽  
Author(s):  
I. M. H. Etherington
Keyword(s):  
Population Genetics ◽  
Linear Algebra ◽  
Point Of View ◽  
Simple Type ◽  
Baric Algebras

Two classes of linear algebras, generally non-associative, are defined in § 3 (baric algebras) and § 4 (train algebras), and the process of duplication of a linear algebra in § 5. These concepts, which will be discussed more fully elsewhere, arise naturally in the symbolism of genetics, as shown in §§ 6–15. Many of their properties express facts well known in genetics; and the processes of calculation which are fundamental in many problems of population genetics can be expressed as manipulations in the genetic algebras. In cases where inheritance is of a simple type (e.g. §§ 10–13, 15) this constitutes a new point of view, but perhaps amounts to little more than a change of notation as compared with existing methods. §14, however, indicates the possibility of generalisations which would seem to be impossible by ordinary methods.

scholarly journals Generalized Standard Triples for Algebraic Linearizations of Matrix Polynomials

2021 ◽  
Vol 37 ◽  
pp. 640-658
Author(s):  
Eunice Y.S. Chan ◽  
Robert M. Corless ◽  
Leili Rafiee Sevyeri
Keyword(s):  
Orthogonal Polynomial ◽  
Matrix Polynomial ◽  
Polynomial Basis ◽  
Regular Matrix ◽  
Bernstein Basis ◽  
Monomial Basis ◽  
Matrix Polynomials ◽  
Regular Matrix Polynomial ◽  
Polynomial Bases

We define generalized standard triples $\boldsymbol{X}$, $\boldsymbol{Y}$, and $L(z) = z\boldsymbol{C}_{1} - \boldsymbol{C}_{0}$, where $L(z)$ is a linearization of a regular matrix polynomial $\boldsymbol{P}(z) \in \mathbb{C}^{n \times n}[z]$, in order to use the representation $\boldsymbol{X}(z \boldsymbol{C}_{1}~-~\boldsymbol{C}_{0})^{-1}\boldsymbol{Y}~=~\boldsymbol{P}^{-1}(z)$ which holds except when $z$ is an eigenvalue of $\boldsymbol{P}$. This representation can be used in constructing so-called  algebraic linearizations for matrix polynomials of the form $\boldsymbol{H}(z) = z \boldsymbol{A}(z)\boldsymbol{B}(z) + \boldsymbol{C} \in \mathbb{C}^{n \times n}[z]$ from generalized standard triples of $\boldsymbol{A}(z)$ and $\boldsymbol{B}(z)$. This can be done even if $\boldsymbol{A}(z)$ and $\boldsymbol{B}(z)$ are expressed in differing polynomial bases. Our main theorem is that $\boldsymbol{X}$ can be expressed using the coefficients of the expression $1 = \sum_{k=0}^\ell e_k \phi_k(z)$ in terms of the relevant polynomial basis. For convenience, we tabulate generalized standard triples for orthogonal polynomial bases, the monomial basis, and Newton interpolational bases; for the Bernstein basis; for Lagrange interpolational bases; and for Hermite interpolational bases.

scholarly journals Polynomial sequences generated by infinite Hessenberg matrices

2017 ◽  
Vol 5 (1) ◽  
pp. 64-72 ◽  
Author(s):  
Luis Verde-Star
Keyword(s):  
Orthogonal Polynomial ◽  
Generating Functions ◽  
Recurrence Relations ◽  
Hessenberg Matrix ◽  
Polynomial Sequences ◽  
Lower Triangular ◽  
Companion Matrices ◽  
Matrix Similarity ◽  
Invertible Matrices ◽  
Construction Algorithms

Abstract We show that an infinite lower Hessenberg matrix generates polynomial sequences that correspond to the rows of infinite lower triangular invertible matrices. Orthogonal polynomial sequences are obtained when the Hessenberg matrix is tridiagonal. We study properties of the polynomial sequences and their corresponding matrices which are related to recurrence relations, companion matrices, matrix similarity, construction algorithms, and generating functions. When the Hessenberg matrix is also Toeplitz the polynomial sequences turn out to be of interpolatory type and we obtain additional results. For example, we show that every nonderogative finite square matrix is similar to a unique Toeplitz-Hessenberg matrix.

A new kind of companion matrix

2017 ◽  
Vol 32 ◽  
pp. 335-342 ◽  
Author(s):  
Eunice Chan ◽  
Robert Corless
Keyword(s):  
Companion Matrix ◽  
Companion Matrices

A new kind of companion matrix is introduced, for polynomials of the form c(λ) = λa(λ)b(λ)+c_0, where upper Hessenberg companions are known for the polynomials a(λ) and b(λ). This construction can generate companion matrices with smaller entries than the Fiedler or Frobenius forms. This generalizes Piers Lawrence’s Mandelbrot companion matrix. The construction was motivated by use of Narayana-Mandelbrot polynomials, which are also new to this paper.

Sparse spectrally arbitrary patterns

2015 ◽  
Vol 28 ◽  
Author(s):  
Brydon Eastman ◽  
Bryan Shader ◽  
Kevin Vander Meulen
Keyword(s):  
Characteristic Polynomial ◽  
Companion Matrix ◽  
Spectrally Arbitrary

We explore combinatorial matrix patterns of order n for which some matrix entries are necessarily nonzero, some entries are zero, and some are arbitrary. In particular, we are interested in when the pattern allows any monic characteristic polynomial with real coefficients, that is, when the pattern is spectrally arbitrary. We describe some order n patterns that are spectrally arbitrary. We show that each superpattern of a sparse companion matrix pattern is spectrally arbitrary. We determine all the minimal spectrally arbitrary patterns of order 2 and 3. Finally, we demonstrate that there exist spectrally arbitrary patterns for which the nilpotent-Jacobian method fails.

Assignment of the characteristic polynomial of a Hessenberg matrix†

1986 ◽  
Vol 44 (1) ◽  
pp. 245-249 ◽  
Author(s):  
S. BARNETT ◽  
C. L. COX ◽  
C. R. JOHNSON
Keyword(s):  
Characteristic Polynomial ◽  
Hessenberg Matrix

A Lagrange series approach to the spectrum of the Kite

2015 ◽  
Vol 30 ◽  
pp. 934-943
Author(s):  
Piet Van Mieghem
Keyword(s):  
Spectral Radius ◽  
Linear Algebra ◽  
Characteristic Polynomial ◽  
Clique Number ◽  
Electronic Journal ◽  
Largest Eigenvalue ◽  
Eigenvalues Of Graphs ◽  
Lagrange Series

A Lagrange series around adjustable expansion points to compute the eigenvalues of graphs, whose characteristic polynomial is analytically known, is presented. The computations for the kite graph P_nK_m, whose largest eigenvalue was studied by Stevanovic and Hansen [D. Stevanovic and P. Hansen. The minimum spectral radius of graphs with a given clique number. Electronic Journal of Linear Algebra, 17:110–117, 2008.], are illustrated. It is found that the first term in the Lagrange series already leads to a better approximation than previously published bounds.

scholarly journals Choosing Improved Initial Values for Polynomial Zerofinding in Extended Newbery Method to Obtain Convergence

2012 ◽  
Vol 2012 ◽  
pp. 1-12 ◽  
Author(s):  
Saeid Saidanlu ◽  
Nor’aini Aris ◽  
Ali Abd Rahman
Keyword(s):  
Characteristic Polynomial ◽  
Initial Approximation ◽  
Companion Matrix ◽  
Good Convergence ◽  
Empirical Results ◽  
Initial Values ◽  
Complex Roots ◽  
Real Value ◽  
Roots Of Polynomials ◽  
Matrix Construction

In all polynomial zerofinding algorithms, a good convergence requires a very good initial approximation of the exact roots. The objective of the work is to study the conditions for determining the initial approximations for an iterative matrix zerofinding method. The investigation is based on the Newbery's matrix construction which is similar to Fiedler's construction associated with a characteristic polynomial. To ensure that convergence to both the real and complex roots of polynomials can be attained, three methods are employed. It is found that the initial values for the Fiedler's companion matrix which is supplied by the Schmeisser's method give a better approximation to the solution in comparison to when working on these values using the Schmeisser's construction towards finding the solutions. In addition, empirical results suggest that a good convergence can still be attained when an initial approximation for the polynomial root is selected away from its real value while other approximations should be sufficiently close to their real values. Tables and figures on the errors that resulted from the implementation of the method are also given.

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