Certain positive definite submatrices that arise from binomial coefficient matrices Original Research Article

We are concerned with the solution of an overdetermined system of compatible linear equations whose matrix arises from binomial coefficients with alternating signs, that is the coefficients in the expansion of (1−1)m. Specifically, we address the problem of choosing a subset of these equations so that the resulting square system is solved efficiently. We find that an enlargement of the given equations establishes a Toeplitz band structure and suggests the choice of a Toeplitz square system whose solution includes the required one. Then we prove some lemmas to justify that the so derived coefficient matrix is positive definite, both for even and odd m. Furthermore, it is shown that the choice of a subsystem may be associated with a principal submatrix of the Toeplitz matrix, thus obtaining positive definiteness. Both the choices allow fast and efficient solvers to be applied.