By Engstrom H. T.
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Additional resources for Periodicity in Sequences Defined by Linear Recurrence Relations
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  Bailey D.
References  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  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)).