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.     

Hankel matrices for system identification

The coefficients of a linear system, even if it is a part of a block-oriented nonlinear system, normally satisfy some linear algebraic equations via Hankel matrices composed of impulse responses or correlation functions. In order to determine or to estimate the coefficients of a linear system it is important to require the associated Hankel matrix be of row-full-rank. The paper first discusses the equivalent conditions for identifiability of the system. Then, it is shown that the row-full-rank of the Hankel matrix composed of impulse responses is equivalent to identifiability of the system. Finally, for the row-full-rank of the Hankel matrix composed of correlation functions, the necessary and sufficient conditions are presented, which appear slightly stronger than the identifiability condition. In comparison with existing results, here the minimum phase condition is no longer required for the case where the dimension of the system input and output is the same, though the paper does not make such a dimensional restriction.     

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.     

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.     

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.     

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.     

Efficient algorithm for Toeplitz plus Hankel matrices

The work of both authors was supported in part by the NSF grant DMS-8801961     

Fast algorithms for Toeplitz and Hankel matrices

The paper gives a self-contained survey of fast algorithms for solving linear systems of equations with Toeplitz or Hankel coefficient matrices. It is written in the style of a textbook. Algorithms of Levinson-type and Schur-type are discussed. Their connections with triangular factorizations, Padè recursions and Lanczos methods are demonstrated. In the case in which the matrices possess additional symmetry properties, split algorithms are designed and their relations to butterfly factorizations are developed.     

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.     

Hankel matrices and quadratic forms

A characterization of finite Hankei matrices is given and it is shown that such matrices arise naturally as matrix representations of scaled trace forms of field extensions and etale algebras. An algorithm is given for calculating the signature and the Hasse invariant of these scaled trace forms.     

Condition numbers of Hankel matrices for exponential weights

We obtain the rate of growth of the largest eigenvalues and Euclidean condition numbers of the Hankel matrices for a general class of even exponential weights W²=exp(−2Q) on an interval I. As particular examples, we discuss Q(x)=α|x| on I=R, and Q(x)=(d²−x²)^−α on I=[−d,d].     

Smallest eigenvalues of Hankel matrices for exponential weights

We obtain the rate of decay of the smallest eigenvalue of the Hankel matrices

     

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.     

How bad are Hankel matrices?

Summary. Considered are Hankel, Vandermonde, and Krylov basis matrices. It is proved that for any real positive definite Hankel matrix of order , its spectral condition number is bounded from below by . Also proved is that the spectral condition number of a Krylov basis matrix is bounded from below by . For , a Vandermonde matrix with arbitrary but pairwise distinct nodes , we show that ; if either or for all , then . Received January 24, 1993/Revised version received July 19, 1993