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Inverses and determinants of Toeplitz-Hessenberg matrices. (English) Zbl 1403.15021

Summary: The inverses of Toeplitz-Hessenberg matrices are investigated. It is known that each inverse of such a matrix is a sum of a lower triangular matrix \(L\) and a matrix \(R\) of rank 1. The formulas of \(L\) and \(x\), \(y\) such that \(xy^T = R\) are derived. Using this result we propose an algorithm for inverting Toeplitz-Hessenberg matrices. Moreover, from the expression of the inverse a formula for the determinant is deduced.

MSC:

15B05 Toeplitz, Cauchy, and related matrices
15A09 Theory of matrix inversion and generalized inverses
15A15 Determinants, permanents, traces, other special matrix functions
65F05 Direct numerical methods for linear systems and matrix inversion
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References:

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