A method for computing all values λ such that A+λB has a multiple eigenvalue

Given a pair of n×n matrices A and B, we consider the problem of finding values λ such that the matrix A+λB has a multiple eigenvalue. Our approach solves the problem using only the standard matrix computation tools. By formulating the problem as a singular two-parameter eigenvalue problem, we construct matrices Δ1 and Δ0 of size 3n2×3n2 with the property that the finite regular eigenvalues of the singular pencil Δ1−λΔ0 are the values λ such that A+λB has a multiple eigenvalue. We show that these values can be computed numerically from Δ1 and Δ0 by the staircase algorithm.