I'm interested in calculating the eigenvalues for the sum of two matrices. I've read several papers on bounds, but I was wondering if anyone knows of a method to find the eigenvalues for the following specific case:

Let $G$ be an $n \times n$ symmetric matrix whose eigenvalues and vectors are known and $A = [1]_{ij}$ (the $n\times n$ matrix of all ones).

Is there any method to find the eigenvalues of the sum $G+A$?

Any help would be greatly appreciated.

  • $\begingroup$ $A$ can be written as $uu^T$ where $u$ is the column vector of all ones. Consequently $G+A$ is a rank-one update of $G$, so tools like the matrix determinant lemma come into play. $\endgroup$ – Semiclassical May 20 '16 at 6:51
  • $\begingroup$ Thanks for the tip. That was helpful. I'll see if I can figure something out. $\endgroup$ – John May 20 '16 at 7:54

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