Eigenvector properties of Toeplitz matrices and their application to spectral analysis of time series

Reddi, S. S.

doi:10.1016/0165-1684(84)90023-9

Cited by 22 publications

(9 citation statements)

References 4 publications

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“…, n}. Then by (15) and the induction hypothesis: ϕ j = ϕ j + ψ 0 = ϕ j−1 + ψ 1 = ( j − 1)ζ + ζ = jζ and ψ k = ψ k + ϕ 0 = ψ k−1 + ϕ 1 = (k − 1)ζ + ζ = kζ . This finishes the induction and we therefore proved | p| = p(ζ x), |q| = q(ζ x).…”

Section: Theorem 12mentioning

confidence: 94%

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Minimizing and maximizing the Euclidean norm of the product of two polynomials

Bünger

2010

Adv Comput Math

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We consider the problem of minimizing or maximizing the quotient f m,n ( p, q) := pq p q ,, Ã ∈ {Ê, }, are non-zero real or complex polynomials of maximum degree m, n ∈ AE respectively and p := (| p 0 | 2 + · · · + |p m | 2 ) 1 2 is simply the Euclidean norm of the polynomial coefficients. Clearly f m,n is bounded and assumes its maximum and minimum values min f m,n = f m,n ( p min , q min ) and max f m,n = f ( p max , q max ). We prove that minimizers p min , q min for Ã = and maximizers p max , q max for arbitrary Ã fulfill deg( p min ) = m = deg( p max ), deg(q min ) = n = deg(q max ) and all roots of p min , q min , p max , q max have modulus one and are simple. For Ã = Ê we can only prove the existence of minimizers p min , q min of full degree m and n respectively having roots of modulus one. These results are obtained by transferring the optimization problem to that of determining extremal eigenvalues and corresponding eigenvectors of autocorrelation Toeplitz matrices. By the way we give lower bounds for min f m,n for real polynomials which are slightly better than the known ones and inclusions for max f m,n .

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Section: Theorem 12mentioning

confidence: 94%

“…In connection with Theorem 9 it immediately implies Corollary 1. For further investigations on roots of eigenvectors of Hermitian Toeplitz matrices we refer to [7][8][9] (survey article), [11,[13][14][15] and [16], Section 9.5, pp. 269-272.…”

Section: Corollary 3 (Carathéodory-representations Of Hermitian Toeplmentioning

confidence: 99%

Minimizing and maximizing the Euclidean norm of the product of two polynomials

Bünger

2010

Adv Comput Math

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“…From (2)(3)(4)(5)(6)(7)(8) we see that = 0. But it also follows from (2-5) that aL+, = 0, and the next reflection coefficient is not determined.…”

Section: I'mentioning

confidence: 94%

Some properties of Hermitian Toeplitz matrices in the singular case

Benidir

Picinbono

1986

1986 25th IEEE Conference on Decision and Control

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“…Also, the location of the zeros of the filters whose coefficients are the elements of the eigenvectors corresponding to the minimum or maximum eigenvalues lie on the unit circle. Moreover as S. S. Reddi (1984) proves these zeros are distinct. Hence it follows that in the noisefree case for infinite length measurements the minimal energy input 5/*f is either symmetric or skew-symmetric with respect to middle.…”

Section: An Overview Of the Identification Frameworkmentioning

confidence: 99%