On a hierarchy of generalized diagonal dominance properties for complex matrices
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- by J. L. Brenner and W. G. Brown
- Proc. Amer. Math. Soc. 25 (1970), 906-911
- DOI: https://doi.org/10.1090/S0002-9939-1970-0260766-1
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Abstract:
This article concerns dominance conditions for an $n \times m$ matrix. In the simplest kind of dominance, the (absolute) value of the diagonal element exceeds the sum of the absolute values of the nondiagonal elements on the same row. This condition has been generalized in the literature in several ways, of which we consider ways in which the rows of the matrix cooperate. Our work amounts to a sorting out of certain dominance conditions that belong to a class $\mathcal {C}$ of dominance conditions. We prove a theorem characterizing all true statements of the form \[ {C_1},{C_2}, \cdots ,{C_s} \Rightarrow {C_0}\] where ${C_i} \in \mathcal {C}(i = 0,1, \cdots ,s)$.References
- J. L. Brenner, Relations among the minors of a matrix with dominant principal diagonal, Duke Math. J. 26 (1959), 563–567. MR 110722
- V. L. Klee Jr., Separation properties of convex cones, Proc. Amer. Math. Soc. 6 (1955), 313–318. MR 68113, DOI 10.1090/S0002-9939-1955-0068113-7
- H. W. Turnbull, The theory of determinants, matrices, and invariants, Dover Publications, Inc., New York, 1960. 3rd ed. MR 0130257
Bibliographic Information
- © Copyright 1970 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 25 (1970), 906-911
- MSC: Primary 15.58
- DOI: https://doi.org/10.1090/S0002-9939-1970-0260766-1
- MathSciNet review: 0260766