There is an extensive body of literature on estimating the eigenvalues of the sum of two symmetric matrices, P + Q, in relation to the eigenvalues of P and Q. Recently, the authors introduced two novel lower bounds on the minimum eigenvalue, λmin(P +Q), under the conditions that matrices P and Q are symmetric positive semi-definite (PSD) and their sum P + Q is non-singular. These bounds rely on the Friedrichs angle between the range spaces of matrices P and Q, which are denoted by R(P) and R(Q), respectively. In addition, both results led to the derivation of several new lower bounds on the minimum singular value of full-rank matrices. We extend these insights to estimate the minimum positive eigenvalue of P + Q, λmin(P + Q), even if P + Q is singular, in terms of the minimum positive eigenvalues of P and Q, namely λmin(P) and λmin(Q). Our approach leverages
angles between specific subspaces of R(P) and R(Q), meticulously chosen to yield a positive lower bound. Additionally, we illustrate these concepts through relevant examples. Finally, we extend our results to complex Hermitian PSD matrices, their convex combinations, and to negative semi-definite matrices.
This is joint work with S. Kirkland (Manitoba) and S. H. Lui (Manitoba).

Refreshments will be served preceding the talk, beginning at 2:45.