A generalized Hertz-type approach to the eigenvalue bounds of complex interval matrices

The paper proposes a methodology for estimating the eigenvalue bounds of complex interval matrices. The theoretical development is based on the use of matrix measures induced by arbitrary absolute and monotone vector norms, as well as the weighting of the matrix norms (implicitly of the matrix measures) by diagonal matrices with positive entries. Thus, for each norm and each diagonal matrix, one obtains a set of bounds (lower and upper for the real and for the imaginary part) expressed as sums of weighted matrix measures. The practical approach focuses on the 2-norm and, for each sum mentioned above, the optimal values of the weighted measures are looked for. The numerical tractability is ensured by global optimization; in our research we have used the MathWorks “ga” solver. By taking equal weights in our practical approach we get tighter bounds than the ones formulated by Hertz (see References), and we therefore consider that our framework generalizes Hertz's. Finally we illustrate the proposed methodology by several examples that yield relevant comparisons with Hertz's procedure and with another technique recently published.