Consider the following block matrix


$$A = \left(\begin{matrix} B & T\\ T & 0 \end{matrix} \right)$$


where all submatrices are square and


* matrix $B = \mbox{diag}\left(b\_1 ,0,0,\dots,0,b\_n \right)$ with $b\_1, b\_n > 0$.
* matrix $T$ is self-adjoint and positive semidefinite.


What can one say about the lowest eigenvalue of this matrix $A$? In particular, how does it depend on the spectrum of $T$ and the entries of $B$? Are there any known results? 



Spectrum of this block matrix
If $\lambda\_\max$ is the greatest eigenvalue of $T$, the least eigenvalue of $A$ is between $-\lambda\_\max$ and $\max(b\_1, b\_n) - \lambda\_\max$. 


