On the reconstruction of Toeplitz matrix inverses from columns (Q1611884)

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 \(T\)?