A symmetric matrix $A$ always has real eigenvalues. If I know the elements on the diagonals, is it possible to have a lower bound on the smallest eigenvalue? How sharp would this bound be?
For now I only found a paper of Hemy WoIkowicz and George P. H. Styan titled as "Bounds for Elgenvalues Using Traces", however, their bounds require the trace of $A^2$ which needs the other entries.
Is there any other bounds or references on this topic?
p.s. I cannot assume that $A$ is positive-definite coz I know that the smallest eigenvalue is $0$ indeed.
Best Answer
No, look at the following matrix and its generalizations $$ \begin{bmatrix}0&a\\a&0\end{bmatrix}$$ Its eigenvalues are $\pm a$ while the diagonal is always zero.