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I'm not sure if this type of matrix has a name, but I feel as if there's a trick to finding the eigenvalues that i'm missing:

$$ a \in R $$ $$ M = \begin{bmatrix} 1 + a & 1 & 1 \\[0.3em] 1 & 1 + a & 1 \\[0.3em] 1 & 1 & 1+ a \end{bmatrix}$$

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marked as duplicate by Davide Giraudo, Hakim, Etienne, vonbrand, Asaf Karagila May 23 '14 at 18:38

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Do you know the eigenvalues in the case $a = 0$?

Do you know the eigenvalues of $A + a \, I$, where $I$ is the identity in terms of the eigenvalues of $A$?

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  • $\begingroup$ Am I right in saying for any matrix (n by n) of ones, the eigenvalues are n (once) and 0 (n-1 times). Then the eigenvalues of A + aI are lambda + am where lambda are the eigenvalues of A? $\endgroup$ – Brian May 23 '14 at 16:46
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The eigenvalues of $M$ are $a+3, \ a, \ a$.

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