A closer look at some new lower bounds on the minimum singular value of a matrix

Abstract

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 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. One significant aspect of the two novel lower bounds on λmin(P+Q) is the distinction of the case where R(P) and R(Q) have no principal angles between 0 and π2. This work offers an explanation for the aforementioned scenario and presents a classification of all matrices that meet the specified criteria. Additionally, we offer insight into the rationale behind selecting the decomposition for the subspace R(Q), which is employed to formulate the lower bounds for λmin(P+Q). At last, an example that showcases the potential for improving these two lower bounds is presented.