On Non-negative-valued Matrices.

cording to our Theorem, if (4) is fulfilled, f3 and a can be so chosen that (2) be fulfilled. This form shows, however, that the lowest term in the determinant Ioua'+ Owa-11, if considered as a polynomial in E, is {nT Hence, the lowest term in the determinant :u ± EWI is I0K-1Iaean-r, i.e., the coefficient of (n7r in the determinant |u + Ew| does not vanish. Conversely, let us assume that u and w are of dimension n, u of rank r, and the coefficient of nr in u + ew does not vanish. The same holds then for u' = fuaand w' = lwa'and we can choose ,3 and a so that u' = $ua-1 have the form given in (2). Furthermore, the algebraic complement of any element in the first r rows of u' + Cw'J goes to zero at least as (n r Hence, the ratio of such a complement to the determinant |u' + EW'|, i.e., any element in the first r rows of (u' + Ew')-', converges to a finite number (or to 0) as e 0. It follows that the elements in the first r rows of (W' + Ew') -1u' converge to the same numbers to which the corresponding elements of (u' + ew') 1(u' + ew') converge. Hence, as far as the first r rows are concerned