By Engstrom H. T.

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**Sample text**

N) where (iv) |ajj | > Pj Q1− j n 0 ≤ ≤ 1, Qj = |akj | ( Marcus and Minc, 1964, p. 150) . 1 yield new invertibility conditions which improve the mentioned results, when the considered matrices are close to triangular ones. Moreover, they give us estimates for diﬀerent norms of the inverse matrices. 1 allow us to derive additional invertibility conditions in the terms of the Euclidean norm. The material in Chapter 3 is based on the papers (Gil’, 1997), (Gil’, 1998) and (Gil’, 2001). References [1] Bailey D.

References [1] Bailey D. W. and D. E. Crabtree, (1969), Bounds for determinants, Linear Algebra and Its Applications, 2, 303-309. 48 3. A. (1982), Matrices, eigenvalues and directed graphs, Linear and Multilinear Algebra, 11, 143-165 [3] Collatz, L. (1966). Functional Analysis and Numerical Mathematics. Academic press, New York and London. O. (1995), Criteria for invertibility of diagonally dominant matrices, Linear Algebra and Its Applications, 215, 63-93. O. (1998), Topics on a generalization of Gershgorin’s theorem, Linear Algebra and Its Applications, 268, 91-116.

15. 15 33 Notes The quantity g(A) was introduced both by P. I. Gil’ (1979b). 1 was derived in the paper (Gil’, 1979a) in a more general situation and was extended in (Gil’, 1995) (see also (Gil’, 1993b)). Recall that Carleman has derived the inequality n (1 − λ−1 λk (A))exp[λ−1 λk (A)] ≤ Rλ (A) k=1 |λ|exp[1 + N 2 (Aλ−1 )/2], cf. (Dunford, N and Schwartz, 1963, p. 1023). 1 was published in (Gil’, 1993a). It improves Schur’s inequality n |λk (A)|2 ≤ N 2 (A) k=1 and Brown’s inequality n |Im λk (A)|2 ≤ N 2 (AI ) k=1 (see (Marcus and Minc, 1964)).