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On the reconstruction of Toeplitz matrix inverses from columns. (English) Zbl 1002.15002

Let \(T\) be an invertible Toeplitz matrix over a commutative field. It is proved via explicit formulas that \(T^{-1}\) can be recovered uniquely from the first column of \(T^{-1}\) and parts of at most two other particular columns (the location of which depends on \(T)\) of \(T^{-1}\), so that the total number of parameters involved is equal to \(2n-1\), where \(n\times n\) is the size of \(T\). Thus, there is no redundancy in the reconstruction of \(T^{-1}\). The result generalizes many earlier findings.
If \(T\) is, in addition, symmetric or skewsymmetric (and the characteristic of the field is different from 2), then \(T^{-1}\) can be recovered uniquely from one particular column of \(T^{-1}\), and, in the symmetric case, from the knowledge of the character of \(T\). The character is equal to 1 or to \(-1\). For Hermitian Toeplitz matrices over the complex field, a similar result is proved, but now the character is a unimodular complex number.
An open question is posed: Can the inverse of an invertible symmetric or Hermitian Toeplitz matrix \(T\) be always recovered from \(n\) entries of \(T^{-1}\), the location of which may depend on \(T\)?

MSC:

15A09 Theory of matrix inversion and generalized inverses
15B57 Hermitian, skew-Hermitian, and related matrices
65F05 Direct numerical methods for linear systems and matrix inversion
15A29 Inverse problems in linear algebra
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