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 ?

  • 3
    $\begingroup$ Welcome to Stackexchange. You'll find that simple "Here's the statement of my question, solve it for me" posts will be poorly received. What is better is for you to add context (with an edit): What you understand about the problem, what you've tried so far, etc.; something both to show you are part of the learning experience and to help us guide you to the appropriate help. You can consult this link for further guidance. $\endgroup$ – GNUSupporter 8964民主女神 地下教會 Oct 10 '18 at 18:54
  • $\begingroup$ I don't think that the fact that $B$ is diagonal and $C,D$ are symmetric is enough to say anything meaningful about the eigenvalues. Is there anything else you can tell us about the matrix? Could you say a bit about why you're interested in the eigenvalues of this matrix? $\endgroup$ – Ben Grossmann Oct 10 '18 at 18:55
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    $\begingroup$ If you're okay with letting $A=C$, then you are looking at a symmetric banded Toeplitz matrix. You may find the following paper helpful: ramanujan.math.trinity.edu/wtrench/research/papers/… $\endgroup$ – Alex Jones Oct 10 '18 at 19:13

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.

  • $\begingroup$ Absolutely right ..thanks $\endgroup$ – user602672 Oct 11 '18 at 3:34

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