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?