The paper deals with the following two problems of completion of block matrices. 1. Let A(ij), i = or > j-q, be given matrices. Find additional matrices A(ij) such that the block matrix A = (A(ij)), i=1 to n, j=1 to m has norm less than one. 2. Let B(ij), absolute value of (j-i) = or < q, be given matrices. Find additional B(ij) such that the completion B = (B(ij)), i,j=1 to n is positive definite. The analysis is based on a study of certain linear fractional maps. The approach leads to an explicit description of all solutions by linear fractional maps of which the coefficients are given directly in terms of the given data. The maximum entropy principle appears as a corollary of a general formula for the determinant of a completion. Special attention is paid to the Toeplitz case.
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