A Class Of Positive Definite Completion Problem For Partial Positive Definite Toeplitz Matrices

A list of postions in an n x n real matrix (a pattern) is said to have postive definitecompletion if every partial positve matrix that specifies exactly these positions can be completed to a postive difinite matrix,especially,when this sysmmetric postions are exactly in some diagonals,which sequences of diagonals ensure the existence of Toeplitz postive definite completion?In this paper it is shown that a partial Toeplitz pattern has a Toeplize positive definite completion if and only if the diagonals for the specified entries are 0,t,2t,...,pt (in which the main diagonal is numbered 0). This gives a answer to the conjecture posed by C. R. Johnson.