In my studies of probability theory, I have recently come across the following eigenvalue problem as part of my research:
Let $ \Sigma $ be a given positive definite $ d \times d $ matrix, and let $ p > 0 $ be a positive constant, we look at the following block matrix
$ M = \left[ {\begin{array}{cc} \Sigma + pI_d & \Sigma \\ \Sigma & \Sigma \\ \end{array} } \right] $
Note: $ I_d $ is the d-dimensional identity matrix.
My question here: How can I compute the eigenvalues of the block matrix $ M $ supposing only that we know eigenvalues and eigenvectors of $ \Sigma $.
I have tried using known results, and yet the eigenvalues elude me. I thought about using Schur complements, but can't figure that out. Also, I thought maybe a brute force solution would work, yet that failed also. I cannot seem to figure out the eigenvalues, so I would certainly appreciate any help on this. I thank all helpers.
$\endgroup$ 51 Answer
$\begingroup$Since the blocks commute, the block matrix determinant formula says the characteristic polynomial of your matrix is $$\eqalign{ \det(M - \lambda I_{2d}) &= \det\pmatrix{\Sigma + (p-\lambda) I_d & \Sigma\cr \Sigma & \Sigma - \lambda I_d}\cr &= \det((\Sigma + (p-\lambda I_d)) (\Sigma - \lambda I_d) - \Sigma^2)\cr&= \det((p-2\lambda) \Sigma + \lambda (\lambda-p) I_d)}$$ This is $0$ iff $\lambda(p-\lambda)/(p-2\lambda)$ is an eigenvalue of $\Sigma$. That is, corresponding to each eigenvalue $\mu$ of $\Sigma$, we have eigenvalues $\lambda = \mu + p/2 \pm \sqrt{\mu^2 + p^2/4}$ of $M$.
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