Can somebody help me in finding eigenvalues of the symmetric matrix $ \pmatrix{A & B\\ B & C}$?
Here $A$ and $C$ are symmetric matrices of order $n$ and $B$ is a diagonal matrix of order $n$.
What I know: If all four blocks (of same size) of a matrix are diagonal matrices then I know how to determine the eigenvalues, for example pickup $i$-th diagonal entries from each block and make one 2 by 2 matrix (position of entries are same as their blocks) whose eigenvalues are also the eigenvalues of the original.
Thus I am curious to know what if I further advance my matrix by removing the case of all four blocks are diagonals? Also to know is there any role of eigenvalues of blocks $A$ and $C$ in determining the eigenvalues of the origianl ?
Since every real symmetric matrix is orthogonally similar to the said form, your problem is not different from the usual symmetric eigenvalue problem. In fact, if one partitions a generic symmetric matrix into $\pmatrix{A&B\\ B^T&C}$ with four equal-sized subblocks, and $B=USV^T$ is a singular value decomposition, then $\pmatrix{U^T\\ &V^T}\pmatrix{A&B\\ B^T&C}\pmatrix{U\\ &V}=\pmatrix{U^TAU&S\\ S&V^TCV}$. So, your assumption about a diagonal $B$ doesn't add anything new to the problem.