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Characteristics of Hankel matrices. (English) Zbl 0838.15014

It is known that given a singular Hankel matrix \(H\), the kernel of \(H\) consists of all polynomials of the form \(m(x) p(x)\), where \(m\) is an arbitrary polynomial whose degree does not exceed a certain bound. The polynomial \(p\), which is the greatest common divisor of all polynomials in the kernel of \(H\), and the bound \(b\) for the degree of \(m\) are the characteristics of \(H\) referred to in the title.
The author presents an approach to characteristic numbers of Hankel matrices based on applications of the infinite companion matrix. His results are consequences of a strengthening of a classical result of Frobenius. For the definitions, see G. Heinig and K. Rost [Algebraic methods for Toeplitz-like matrices and operators (1984; Zbl 0549.15013)].

MSC:

15B57 Hermitian, skew-Hermitian, and related matrices

Citations:

Zbl 0549.15013
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References:

[1] M. Fiedler: Quasidirect decompositions of Hankel and Toeplitz matrices. Lin. Algebra Appl. 61 (1984), 155-174. · Zbl 0548.15022
[2] M. Fiedler: Polynomials and Hankel matrices. Lin. Algebra Appl. 66 (1985), 235-248. · Zbl 0569.15005
[3] M. Fiedler, V. Pták: Intertwining and testing matrices corresponding to a polynomial. Lin. Algebra Appl. 86 (1978), 53-74. · Zbl 0621.15016
[4] G. Heinig, K. Rost: Algebraic Methods for Toeplitz-like Matrices and Operators. Akademie-Verlag, Berlin, 1984. · Zbl 0549.15013
[5] A.S. Iochvidov: Hankel and Toeplitz Matrices and Forms. Nauka, Moscow, 1974.
[6] V. Pták: The infinite companion matrix. Lin. Algebra Appl. 166 (1992), 65-95. · Zbl 0749.15015
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