Estimation of ‖A−1‖∞‖A−1‖∞ and the smallest singular value

An upper bound for ‖A−1‖∞‖A−1‖∞ and a lower bound for the smallest singular value, for the weakly diagonally dominant matrices we introduce here(which are an extended matrix class of the strictly diagonally dominant matrices), are presented. Examples show that the new bounds improve the bounds in Varah [J.M. Varah, A lower bound for the smallest singular value, Linear Algebra Appl. 11 (1975) 3–5], Johnson [C.R. Johnson, A Gersgorin-type lower bound for the smallest singular value, Linear Algebra Appl. 112 (1989) 1–7] and Johnson, Szulc [C.R. Johnson, T. Szulc, Further lower bounds for the smallest singular value, Linear Algebra Appl. 272 (1998) 169–179].