Reconstruction of a positive definite Toeplitz matrix from its sequence of minimum eigenvalues

It is shown that a positive definite symmetric Toeplitz matrix of rank p+1 can be represented uniquely by the minimum eigenvalues of itself and its p submatrices, along with a sign bit for each eigenvalue. The application of such a procedure allows specification of linear prediction coding (LPC) spectra by an alternate set of parameters. It also enables extension of an autocorrelation sequence using a nonstandard criterion. A number of parallels are made between these parameters and the LPC mean-square error values for successively higher-order systems. An application involving maximum entropy spectral estimation under a nonstandard set of constraints is presented