I have been struggling with this problem for some time and wondering if there exists a theorem related to this.
Given two real symmetric matrices $A\in \mathcal{R}^{N\times N}$ and $B\in \mathcal{R}^{N\times N}$; both of them have equal diagonal values. Is it possible to comment on $\lambda_A$ and $\lambda_B$ which represents the smallest eigenvalues of A and B respectively? Can we say $\lambda_A$ < $\lambda_B$ or otherwise depending on some constraints?
Any hints or suggestions are welcome.