Information requirements for toeplitz inversion
Charles R. Johnson, Tamir Shalom · Linear and Multilinear Algebra · 1991
It is known that there exist sets of n vectors U 1 U 2,…,Un ε such that the invertibility of any n-by-n nonsingular Toeplitz matrix A may be decided, and in this event its inverse reconstructed, from the solutions to AX=U 1 for 3 choices of i (that depend upon A). It is shown here that the inverse of such an A cannot generally be reconstructed from the solutions to AX=U 1 any n−1 fixed vectors U 1 U 2,…,Un−1 ε nor may invertibility even be decided. This situation lies in contrast to another familiar class of structured matrices, the circulants. Other related issues are addressed, and the notion of a Toeplitz basis is introduced.