On normal Hankel matrices of low orders

In the previous work of the authors, the problem of describing complex n × n matrices that are simultaneously normal and Hankel was reduced to a system of n ? 1 real equations with respect to 2n unknowns. These equations are quadratic, and it is not at all clear whether they have real solutions. It is shown here that the systems corresponding to n = 3 and n = 4 are solvable and have infinitely many real solutions.     

On parametrization of totally nonpositive matrices and applications

The problem of accurate computations for totally non‐negative matrices has been studied; however, it remains open for other sign regular matrices. One major obstacle is that there is no known parametrization of these matrices. The main contribution of the present work is that we provide such parametrization of nonsingular totally nonpositive matrices. A useful application of our results is that these parameters can determine accurately the entries of the inverse of a nonsingular totally nonpositive matrix. Copyright © 2011 John Wiley & Sons, Ltd.     

s-Numbers of Hankel matrices

Characterizations are obtained for the s-numbers of a bounded Hankel matrix H = (c_{j + k}), i.e., for the upper enumeration of the eigenvalues of the operator $\text{¦ H ¦ = (H}{}^1\text{H)}{}^{\mathrm{<mtext>1</mtext><mtext>2</mtext>}}$ and for the supremum of its essential spectrum.     

On conditions for permutability of Toeplitz and Hankel matrices

The problem of describing pairs of commuting matrices (T, H), where T and H are a Toeplitz and a Hankel matrix, respectively, is examined. Several families of such pairs are indicated.     

Polynomials and Hankel matrices

Compatibility of a Hankel n × n matrix H and a polynomial f of degree m, m ? n, is defined. If m = n, compatibility means that HC^T_f=C_fH where C_f is t companion matrix of f. With a suitable generalization of C_f, this theorem is generalized to the case that m < n.     

Hankel and Loewner matrices

Mutual relations between the Hankel, Toeplitz, Bézout, and Loewner matrices as well as further connections to rational interpolation and projective geometry are investigated.     

On the problem of describing pairs of commuting Hankel complex matrices

Solutions of two partial problems arising in the course of classifying pairs of commuting Hankel complex matrices are presented.     

Hankel matrices for the period-doubling sequence

We give an explicit evaluation, in terms of products of Jacobsthal numbers, of the Hankel determinants of order a power of two for the period-doubling sequence. We also explicitly give the eigenvalues and eigenvectors of the corresponding Hankel matrices. Similar considerations give the Hankel determinants for other orders.     

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.     

Permanental ideals of Hankel matrices

We provide the Gröbner basis and the primary decomposition of the ideals generated by 2 × 2 permanents of Hankel matrices.     

Commutation conditions for Hankel matrices

Commutation conditions for Hankel matrices are found. As a consequence, the classification of normal Hankel matrices given earlier by Kh.D. Ikramov and V.N. Chugunov is validated.