New lower bounds on the minimum singular value of a matrix

The study of constraining the eigenvalues of the sum of two symmetric matrices, say P + Q, in terms of the eigenvalues of P and Q, has a long tradition. It is closely related to estimating a lower bound on the minimum singular value of a matrix, which has been discussed by a great number of authors. To our knowledge, no study has yielded a positive lower bound on the minimum eigenvalue, Amin(P + Q), when P + Q is symmetric positive definite with P and Q singular positive semi-definite. We derive two new lower bounds on Amin(P + Q) in terms of the minimum positive eigenvalues of P and Q. The bounds take into account geometric information by utilizing the Friedrichs angles between certain subspaces. The basic result is when P and Q are two non-zero singular positive semi-definite matrices such that P+Q is non-singular, then Amin(P + Q) i (1 -cos theta F) min{Amin(P), Amin(Q)}, where Amin represents the minimum positive eigenvalue of the matrix, and theta F is the Friedrichs angle between the range spaces of P and Q. Such estimates lead to new lower bounds on the minimum singular value of full rank 1 x 2, 2 x 1, and 2 x 2 block matrices. Some examples provided in this paper further highlight the simplicity of applying the results in comparison to some existing lower bounds.(c) 2023 Elsevier Inc. All rights reserved.