Toeplitz determinants with perturbations in the corners. (English) Zbl 1300.15010
Summary: The paper is devoted to exact and asymptotic formulas for the determinants of Toeplitz matrices with perturbations by blocks of fixed size in the four corners. If the norms of the inverses of the unperturbed matrices remain bounded as the matrix dimension goes to infinity, then standard perturbation theory yields asymptotic expressions for the perturbed determinants. This premise is not satisfied for matrices generated by so-called Fisher-Hartwig symbols. In that case, we establish formulas for pure single Fisher-Hartwig singularities and for Hermitian matrices induced by general Fisher-Hartwig symbols.
MSC:
| 15B05 | Toeplitz, Cauchy, and related matrices |
| 15A15 | Determinants, permanents, traces, other special matrix functions |
| 15A60 | Norms of matrices, numerical range, applications of functional analysis to matrix theory |
| 47B35 | Toeplitz operators, Hankel operators, Wiener-Hopf operators |
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