On eigenvalues of real symmetric interval matrices: Sharp bounds and disjointness

Abstract

In this paper, the eigenvalue problem of real symmetric interval matrices is studied. First, in the case of $2 \times 2$ real symmetric interval matrices, all the four endpoints of the two eigenvalue intervals are determined. These are not necessarily eigenvalues of vertex matrices, but it is shown that such a real symmetric interval matrix can be constructed from the original one. Then, necessary and sufficient conditions are provided for the disjointness of eigenvalue intervals. In the general $n\times n$ case, due to Hertz, a set of special vertex matrices determines the maximal eigenvalue and a similar statement holds for the minimal one. In a special case, namely if the right endpoints of the off-diagonal intervals are not smaller than the absolute value of the left ones, he concluded the vertex matrix of the right endpoints provides the maximal eigenvalue. Generalizing it, it is shown that in the case of real symmetric interval matrices with special sign pattern, a single vertex matrix determines one of the extremal bounds.