Geodesic
Bounds for the minimum eigenvalue of a symmetric Toeplitz matrix
Electronic transactions on numerical analysis, Tome 8 (1999), pp. 127-137
In a recent paper Melman [12] derived upper bounds for the smallest eigenvalue of a real symmetric Toeplitz matrix in terms of the smallest roots of rational and polynomial approximations of the secular equation $f(\lambda ) = 0$, the best of which being constructed by the (1, 2)-Pad$\acute e$ approximation of f. In this paper we prove that this bound is the smallest eigenvalue of the projection of the given eigenvalue problem onto a Krylov space of T - 1 n of dimension 3. This interpretation of the bound suggests enhanced bounds of increasing accuracy. They can be substantially improved further by exploiting symmetry properties of the principal eigenvector of Tn.
Classification : 65F15
Keywords: Toeplitz matrix, eigenvalue problem, symmetry 
@article{ETNA_1999__8__a2,
     author = {Voss,  Heinrich},
     title = {Bounds for the minimum eigenvalue of a symmetric {Toeplitz} matrix},
     journal = {Electronic transactions on numerical analysis},
     pages = {127--137},
     year = {1999},
     volume = {8},
     zbl = {0936.65044},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/ETNA_1999__8__a2/}
}
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Voss,  Heinrich. Bounds for the minimum eigenvalue of a symmetric Toeplitz matrix. Electronic transactions on numerical analysis, Tome 8 (1999), pp. 127-137. http://geodesic.mathdoc.fr/item/ETNA_1999__8__a2/