Iterative methods for solving partial difference equations of elliptic type
TL;DR: In this paper, the determinant of the matrix A = (ai,j) does not vanish and if A * = (a*j) is symmetric, where a*1=ai,iai,j/ai,i (i, j= 1, 2, N *, N), then A * is positive definite.
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Abstract: Conditions (1.2) were formulated by Geiringer [4, p. 379](2). Evidently these conditions imply that ai,i 5#O (i=1, 2, -, 1N). It is easy to show by methods similar to those used in [4, pp. 379-381] that the determinant of the matrix A= (ai,j) does not vanish. Moreover, if the matrix A * = (a*j) is symmetric, where a*1=ai,iai,j/ ai,i (i, j= 1, 2, N * , N), then A * is positive definite. For if X is a nonpositive real number, then the matrix A * -XI, where I is the identity matrix, also satisfies (1.2) and hence its determinant cannot vanish. Therefore all eigenvalues of A * are positive, and A * is positive definite. On the other hand if A* is positive definite then ai,i5zQ (i=1, 2, , N). We shall be concerned with effective methods for obtaining numerical solu-
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