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, λ min (P + Q) , when P + Q is symmetric positive definite with P and Q singular positive semi-definite. We derive two new lower bounds on λ min (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 λ min (P + Q) ≥ (1 − cos ⁡ θ F) min ⁡ { λ min (P) , λ min (Q) } , where λ min represents the minimum positive eigenvalue of the matrix, and θ 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 × 2 , 2 × 1 , and 2 × 2 block matrices. Some examples provided in this paper further highlight the simplicity of applying the results in comparison to some existing lower bounds. [ABSTRACT FROM AUTHOR]