Dr Vanni Noferini from the Department of Mathematical Sciences
This talk is based on two papers in preparation, coauthored jointly with Yuji Nakatsukasa (Oxford), and will be made of two halves.
The first half is about eigenvalues of matrices whose entries are Lipschitz continuous functions of a real parameter. In the case of Hermitian matrices, if the eigenvalues of such a matrix H(t) are plotted against t, it can typically be observed that the trajectories of the eigenvalue functions appear to collide; however, they undergo a last-minute repulsive effect, thus avoiding intersections .This phenomenon was first explained by Von Neumann and Wigner in 1929. Dr Noferini plans to describe the Neumann-Wigner approach, and then to expose our recent generalization to several different classes of matrices (beyond Hermitian).
The second half is also on a matrix theoretical result, but one that Dr Noferini also expects to have practical relevance for people dealing with the numerical solution of generalized and nonlinear Hermitian eigenvalue problems. Sylvester's law of inertia states that the number of positive, zero or negative eigenvalues of a matrix is invariant under congruence, and the same is true for pencils when at least one matrix is definite (and both are allowed to undergo independent congruences). Nothing was known thus far for indefinite pencils, and almost nothing for nonlinear problems. Dr NoferiniI will present new results in this area, including inertia-based lower and upper bounds for the number of eigenvalues in a real interval.