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

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, ${\lambda }_{min}(P+Q)$, 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 $R\left(P\right)$ and $R\left(Q\right)$, 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 ${\lambda }_{min}(P+Q)$ is the distinction of the case where $R\left(P\right)$ and $R\left(Q\right)$ have no principal angles between 0 and $\frac{\pi }{2}$. 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 $R\left(Q\right)$, which is employed to formulate the lower bounds for ${\lambda }_{min}(P+Q)$. At last, an example that showcases the potential for improving these two lower bounds is presented.