Eigenvalue Bound of Block Matrices

Question

I have the following eigenvalue problem for block matrices A and B

\begin{equation} \left[ \begin{array}{ccc} A_{11} & A_{12} & A_{13} \\ A_{21} & A_{22} & A_{23} \\ A_{31} & A_{32} & A_{33} \end{array} \right]v = \lambda \left[ \begin{array}{ccc} B_{11} & B_{12} & 0 \\ B_{21} & B_{22} & 0 \\ 0 & 0 & B_{33} \end{array} \right]v \end{equation} where matrix $B$ is symmetric and positive definite and matrix $A$ is symmetric and positive semi-definite. All blocks are square and of the same size.

I) If $B_{12}=B_{21}=0$, Can I say anything about $\max(\lambda)$?

II) How does setting $B_{12}$ and $B_{21}$ to zero shift $\max(\lambda)$?

Answer 1

Hint: if $$\begin{bmatrix} A_{11} & A_{12} & A_{13} \end{bmatrix} \begin{bmatrix} v_1 \\ v_2 \\ v_3 \end{bmatrix}=\lambda B_{11}v_1$$ for case (I), then the first entry works.