On the optimality and sharpness of Laguerre's lower bound on the smallest eigenvalue of a symmetric positive definite matrix

Lower bounds on the smallest eigenvalue of a symmetric positive definite matrices $A\in {\mathbb{R}}^{m\times m}$ play an important role in condition number estimation and in iterative methods for singular value computation. In particular, the bounds based on $\mathrm{T}\mathrm{r}(A^{-1})$ and $\mathrm{T}\mathrm{r}(A^{-2})$ attract attention recently because they can be computed in $O(m)$ work when $A$ is tridiagonal. In this paper, we focus on these bounds and investigate their properties in detail. First, we consider the problem of finding the optimal bound that can be computed solely from $\mathrm{T}\mathrm{r}(A^{-1})$ and $\mathrm{T}\mathrm{r}(A^{-2})$ and show that so called Laguerre’s lower bound is the optimal one in terms of sharpness. Next, we study the gap between the Laguerre bound and the smallest eigenvalue. We characterize the situation in which the gap becomes largest in terms of the eigenvalue distribution of $A$ and show that the gap becomes smallest when $\mathrm{T}\mathrm{r}(A^{-2})\mathrm{/}\{\mathrm{T}\mathrm{r}(A^{-1}){\}}^2$ approaches 1 or $\frac{1}{m}$. These results will be useful, for example, in designing efficient shift strategies for singular value computation algorithms.