On product of companion matrices

This paper describes an explicit combinatorial formula for the product of companion matrices. The method relies on the connections between matrix algebra and associated combinatorial structures to enumerate the paths in an unweighted digraph. As an application, we obtain bases for the solution space of the linear difference equation with variable coefficients.     

Applications of doubly companion matrices

Doubly companion matrices are used as a tool to analyze the ESIRK and DESIRE methods, and the general linear methods satisfying the IRKS property. These methods can be considered as extensions of the DIRK and SIRK Runge–Kutta methods, and were introduced to overcome some of the undesirable properties of these methods. A connection between the ESIRK and DESIRE methods, and IRKS methods is also established. Several low order methods are given.     

On compositive functions of matrices

Annali di Matematica Pura ed Applicata (1923 -) -     

Three-way companion matrices of three-variable polynomials

The existence of the companion matrices of three-variable polynomials is investigated. A theorem giving necessary and sufficient conditions for the existence of a three-way companion matrix is stated and proved. The construction of such a matrix is given too. The problem of the factorization of three-variable polynomials is also solved.     

Numerical ranges of reducible companion matrices

In this paper, we show that a reducible companion matrix is completely determined by its numerical range, that is, if two reducible companion matrices have the same numerical range, then they must equal to each other. We also obtain a criterion for a reducible companion matrix to have an elliptic numerical range, put more precisely, we show that the numerical range of an n-by-n reducible companion matrix C is an elliptic disc if and only if C is unitarily equivalent to A⊕B, where A∈M_n-2, B∈M₂ with σ(B)={aω₁,aω₂}, , ω₁≠ω₂, and .     

Some decomposition results for companion matrices

Let C(p) denote the Frobenius companion matrix of the monic polynomial p with complex coefficients. The singular value decomposition and the QR-decomposition of C(p) are computed.     

Hankel matrices and the infinite companion

A formula of Barnett type relating the Bezoutian B(f,g) to the Hankel matrix H(g/f) is extended to rectangular Bezoutians. The proof is based on an interesting relation between the family of all Hankel matrices corresponding to the Markov parameters of g/f and the infinite companion matrix corresponding to f.     

On the calculation of functions of matrices

The representation of entire functions of matrices via symmetric polynomials of nth order is obtained. A method of deriving analytic formulas for functions of matrices of second, third, and fourth orders is obtained. Symmetric polynomials are used to construct algorithms for the numerical calculations of entire functions of matrices, in particular, of matrix exponentials, not requiring the determination of the eigenvalues of the matrices. The efficiency of the proposed numerical methods is estimated.     

Pseudozeros of polynomials and pseudospectra of companion matrices

Summary. It is well known that the zeros of a polynomial are equal to the eigenvalues of the associated companion matrix . In this paper we take a geometric view of the conditioning of these two problems and of the stability of algorithms for polynomial zerofinding. The is the set of zeros of all polynomials obtained by coefficientwise perturbations of of size ; this is a subset of the complex plane considered earlier by Mosier, and is bounded by a certain generalized lemniscate. The is another subset of defined as the set of eigenvalues of matrices with ; it is bounded by a level curve of the resolvent of $A$. We find that if $A$ is first balanced in the usual EISPACK sense, then and are usually quite close to one another. It follows that the Matlab ROOTS algorithm of balancing the companion matrix, then computing its eigenvalues, is a stable algorithm for polynomial zerofinding. Experimental comparisons with the Jenkins-Traub (IMSL) and Madsen-Reid (Harwell) Fortran codes confirm that these three algorithms have roughly similar stability properties. Received June 15, 1993     

Subalgebras of matrix algebras generated by companion matrices

Let f,g∈Z[X] be monic polynomials of degree n and let C,D∈M_n(Z) be the corresponding companion matrices. We find necessary and sufficient conditions for the subalgebra Z〈C,D〉 to be a sublattice of finite index in the full integral lattice M_n(Z), in which case we compute the exact value of this index in terms of the resultant of f and g. If R is a commutative ring with identity we determine when R〈C,D〉=M_n(R), in which case a presentation for M_n(R) in terms of C and D is given.     

Rings of matrices generated by a companion matrix

The regular representation of the quotient of a polynomial ring over the principal ideal determined by h(x) is the ring of matrices generated by the companion matrix of h(x). Properties of such rings, also called Barnett matrix rings, will be investigated.     

A block bidiangonal form for block companion matrices

The problem of the linear factorization of a polynomial matrix is related with a similarity condition linking the block companion matrix and a block upper bidiagonal matrix constructed from a chain of solvents. This result is the applied to the solution of differential and difference linear matrix equations.     

Resultants,companion matrices,and Jacobson chains

The multiplicative property of triangular Toeplitz matrices is used to derive a polyproduct rule for companion matrices. This rule is then used to obtain some new identities involving resultants and companion matrices, and to investigate the block structure of Jacobson chains relating companion and hypercompanion matrices.     

On functions which preserve the class of Stieltjes matrices

A positive definite symmetric matrix is called a Stieltjes matrix provided that all its off diagonal elements are nonpositive. We characterize functions which preserve the class of Stieltjes matrices.     

A fast implicit QR eigenvalue algorithm for companion matrices

An implicit version of the shifted QR eigenvalue algorithm given in Bini et al. [D.A. Bini, Y. Eidelman, I. Gohberg, L. Gemignani, SIAM J. Matrix Anal. Appl. 29(2) (2007) 566-585] is presented for computing the eigenvalues of an n×n companion matrix using O(n²) flops and O(n) memory storage. Numerical experiments and comparisons confirm the effectiveness and the stability of the proposed method.     

A noncommutative spectral theorem for operator entried companion matrices

The scalar entries of the classical companion matrix are replaced by certain operator polynomials defined on an infinite dimensional Hilbert space and the spectrum of the resulting operator entried matrix is determined.     

On infinitely divisible matrices,kernels, and functions

Summary Generalizations and applications of a recent theorem of C. Loewner on nonnegative quadratic forms have led to interesting new results and to some especially simple derivations of well-known theorems from a unified point of view. In this note we discuss two applications to the study of characteristic functions and completely monotonic functions, and show how the classical representation theorems for infinitely divisible laws may be obtained.Research partially sponsored by National Science Foundation under Grants GP 5855 and GP 5358