Regularity radius and real eigenvalue range

Let A(p)A(p) be a given (n×n)(n×n) interval parametric matrix (the set of all (n×n)(n×n) parametric matrices when the m-dimensional parameter vector p   varies within a given interval vector pp) whose entries depend affine linearly on p. Also, let L   denote the set of all real eigenvalues of the bundle (A(p),B(p))(A(p),B(p)). In this paper, first the concept of regularity radius r∗(A(p))r∗(A(p)) of the interval parametric matrix A(p)A(p) is introduced. It is then shown that there exists certain relationship between the problem of establishing if α∈Rα∈R belongs to L   or not and the numerical value of the regularity radius of the interval parametric matrix A(p)-αB(p)A(p)-αB(p). The results presented may be useful in designing a method for determining or assessing the set L.