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Novel estimations for the eigenvalue bounds of complex interval matrices

Applied Mathematics and Computation
Publication Date
DOI: 10.1016/j.amc.2014.02.054
  • Interval Matrix
  • Eigenvalue Bounds
  • Inequalities Involving Matrices
  • Inequalities Involving Eigenvalues
  • Computer Science
  • Mathematics


Abstract Our work proposes two methods that estimate the eigenvalue bounds (left/right for real and imaginary parts) of complex interval matrices. The first method expresses each bound as an algebraic sum of weighted matrix measures, where the measure corresponds to the spectral norm and the weighting matrix is diagonal and positive definite, with unknown entries. The optimization with respect to the entries of the weighting matrix yields the best value of the bound; the computational approach is ensured as the minimization of a linear objective function subject to bilinear-matrix-inequality constraints and interval constraints. The bounds are proved to be better than those provided by other estimation techniques. The second method constructs four real matrices so that each of them can be exploited as a comparison matrix for the complex interval matrix and allows the estimation of one of the eigenvalue bounds. The two methods we propose rely on different mathematical backgrounds, and between the resulting bounds no firm inequality can be stated; therefore, they are equally useful in applications, as reflected by the numerical case studies presented in the paper.

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