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Stabilization by a diagonal matrix

Author: C. S. Ballantine
Journal: Proc. Amer. Math. Soc. 25 (1970), 728-734
MSC: Primary 15.25
MathSciNet review: 0260765
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Abstract: In this paper it is shown that, given a complex square matrix $ A$ all of whose leading principal minors are nonzero, there is a diagonal matrix $ D$ such that the product $ DA$ of the two matrices has all its characteristic roots positive and simple. This result is already known for real $ A$, but two new proofs for this case are given here.

References [Enhancements On Off] (What's this?)

  • [1] K. Fan, Topological proofs for certain theorems on matrices with non-negative elements, Monatsh. Math. 62 (1958), 219-237. MR 20 #2354. MR 0095856 (20:2354)
  • [2] M. E. Fisher and A. T. Fuller, On the stabilization of matrices and the convergence of linear iterative processes, Proc. Cambridge Philos. Soc. 54 (1958), 417-425. MR 20 #2086. MR 0095584 (20:2086)

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Keywords: Diagonal matrix, stable matrix, leading principal minors, characteristic roots, positive simple roots, continuity of roots, separation of roots, parallelotope, continuous mapping
Article copyright: © Copyright 1970 American Mathematical Society

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