Using the duality relations associated with Gauss's method it is shown that: i. Any main submatrix of a 'totally non-degenerate matrix A is stipulated (in the sense of an arbitrary monotonic norm) to be not inferior to A itself; 2. If the inverse of matrix A is diagonally dominant with respect to the columns, then the linear system Ax=b is regularly solved by Jordan's method without the main element being chosen. i. Throughout, A denotes a square non-degenerate matrix of order n with non-zero corner minors, and Ban inverse of A.
We distinguish between the concept of a main submatrix, that is of the arbitrary submatrix located on the intersection of groups of rows and columns with the same numbers, and the corner submatrix as a main submatrix located at either the left upper or the right lower corner of a given matrix (and correspondingly, we speak of upperand lower corner submatrices).
The same applies to the concept of the main and corner minors. We denote by A~(Ak I) the upper (lower) corner submatrix of matrix A, of order k. ~W,, DDR., DDC,, TN~', DDRn -~, DDC,-, denote, in the order indicated, the classes nXn of the nondegenerate totally non-negative matrices, the matrices diagonally predominant with respect to rows, matrices diagonally predominant with respect to columns, and the corresponding classes of the inverse matrices. We note that the matrix norm Ib,!i is referred to as monotonic if for the arbitrary matrices F and G with identical dimensions from l/~JI~lg0l we have iI~[l~ilG~l for any i, ].In particular, the well-known norms Iblh, If.If., If'TiE and the norm IEa[l~=n max I~ol. are monotonic. ~'J 2. The majority of the statements of this paragraph are based on the relation formulated in Lemma i, between the main submatrices of A, and the submatrices of B, obtained by Gaussian elimination (the parts played by A and Bare of course interchangeable).
Assume that the order of rows and columns of A , and therefore of B, is fixed.We denote by B (h' the lower corner n-th order submatrix of the matrix B,_k=(b~ ~) obtained from B as a result of n--k steps of the Gaussian method.