THE EXTREME EIGENVALUES AND STABILITY OF REAL SYMMETRICAL INTERVAL MATRICES

In this note we present a novel Kharitonov-like algorithm to compute the minimal and maximal eigenvalues of n x n dimensional symmetric interval matrices. We prove that the maximal eigenvalue of a given set of interval matrices coincides with the maximal eigenvalue of a special set of 2n-1 symmetric vertex matrices, whereas its minimal eigenvalue coincides with the minimal eigenvalue of another special set of 2n-1 symmetric vertex matrices. As immediate corollaries of this algorithm eigenvalue, we obtain strong necessary and sufficient conditions for testing Hurwitz and Schur stability of symmetric interval matrices, where one has to test stability of 2n-1 and 2n symmetric vertex matrices, respectively. Note that previous weak necessary and sufficient conditions of Soh for testing Hurwitz and Schur stability of symmetric interval matrices require testing stability of all the 2(n2 + n)/2 symmetric vertex matrices. For example, suppose that using a fast computer we can carry out 10(9) almost-equal-to 2(30) Hurwitz stability tests. Then, using the proposed strong conditions one can handle 31 x 31 dimensional matrices, whereas using the weak conditions of Soh one can handle only 6 x 6 dimensional matrices.