Novel estimations for the eigenvalue bounds of complex interval matrices Mihaela-Hanako Matcovschi, Octavian Pastravanu ⇑ Department of Automatic Control and Applied Informatics, Technical University ‘‘Gheorghe Asachi’’ of Iasi, Boulevard Mangeron 27, 700050 Iasi, Romania article info Keywords: Interval matrix Eigenvalue bounds Inequalities involving matrices Inequalities involving eigenvalues 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. Ó 2014 Elsevier Inc. All rights reserved. 1. Introduction Let the real matrices A  ; A þ ; B  ; B þ 2 R nn satisfy the componentwise inequalities A  6 A þ ; B  6 B þ , and define the real interval matrices A¼½A  ; A þ ¼ A 0 þ ½R A ; R A ¼ A 2 R nn j A  6 A 6 A þ    R nn ; ð1Þ B¼½B  ; B þ ¼ B 0 þ ½R B ; R B ¼ B 2 R nn j B  6 B 6 B þ    R nn ð2Þ having A 0 ¼ 1 2 ðA  þ A þ Þ; B 0 ¼ 1 2 ðB  þ B þ Þ as their centers, and the nonnegative matrices R A ¼ 1 2 ðA þ  A  Þ P 0; R B ¼ 1 2 ðB þ  B  Þ P 0 as their radii. The family (set) of complex matrices C¼Aþ jB¼ C ¼ A þ jBj A 2A; B 2B f g C nn ; ð3Þ where j is the imaginary unit, i.e. j 2 ¼1, is called a complex interval matrix. For the description of C we are also going to use the complex matrices C 0 ¼ A 0 þ jB 0 2 C nn and R ¼ R A þ jR B 2 C nn , with A 0 ; B 0 ; R A ; R B 2 R nn introduced above. http://dx.doi.org/10.1016/j.amc.2014.02.054 0096-3003/Ó 2014 Elsevier Inc. All rights reserved. ⇑ Corresponding author. E-mail addresses: mhanako@ac.tuiasi.ro (M.-H. Matcovschi), opastrav@ac.tuiasi.ro (O. Pastravanu). Applied Mathematics and Computation 234 (2014) 645–666 Contents lists available at ScienceDirect Applied Mathematics and Computation journal homepage: www.elsevier.com/locate/amc