Applied and Computational Matrix Analysis: MAT-TRIAD, by Natália Bebiano

By Natália Bebiano

This quantity provides fresh advances within the box of matrix research in keeping with contributions on the MAT-TRIAD 2015 convention. issues lined comprise period linear algebra and computational complexity, Birkhoff polynomial foundation, tensors, graphs, linear pencils, K-theory and statistic inference, displaying the ubiquity of matrices in numerous mathematical areas.

With a specific specialise in matrix and operator concept, statistical types and computation, the foreign convention on Matrix research and its purposes 2015, held in Coimbra, Portugal, used to be the 6th in a chain of meetings.

Applied and Computational Matrix Analysis will attract graduate scholars and researchers in theoretical and utilized arithmetic, physics and engineering who're looking an outline of modern difficulties and techniques in matrix analysis.

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J. Comput. Appl. Math. 212(1), 1–15 (2008) 14. : The question of uniqueness for G. D. Birkhoff interpolation problems. J. Approx. Theory 2, 1–28 (1969) 15. : Applying quantifier elimination to the Birkhoff interpolation problem. J. Symb. Comput. 22(1), 83–103 (1996) 16. : Runge-Kutta defect control using Hermite-Birkhoff interpolation. SIAM J. Sci. Comput. 12, 991–999 (1991) 17. : Birkhoff interpolation problem. Report CNA-103. The Center of Numerical Analysis, The University of Texas (1975) Birkhoff Polynomial Basis 25 18.

I=1 32 M. Hladík To estimate m 1 is a more involved task. For any nonsingular matrix M ∈ Rn×n , its determinant (and also sub-determinant) is bounded by det(M) = det(M −1 )−1 ≥ ρ(M −1 )−n ≥ σmax (M −1 )−n = σmin (M)n . This bound, however, can be very conservative. Anyway, we arrive at the possible value of α := σmin (M)n 2 log 2 · n . 1 Sufficient Conditions for P-Matrices Characterizations of P-matrix property from the previous section enables us to derive new sufficient conditions. Theorem 8 The matrix A ∈ Rn×n is a P-matrix provided A − I and A + I are nonsingular and ρ(|(A + I )−1 (A − I )|) < 1.

Cz © Springer International Publishing AG 2017 N. 1007/978-3-319-49984-0_2 27 28 M. Hladík The set of interval matrices of size m × n is denoted by IRm×n . For definition of interval arithmetic see [8, 10], for instance. We say that A is regular if every A ∈ A is nonsingular. , in [5, 15, 16]. In particular, Rohn [16] presents forty equivalent characterizations. NP-hardness of checking regularity was proven by Poljak and Rohn [12, 13]. Sufficient conditions for checking regularity are surveyed in Rex and Rohn [14].

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