I have an $n\times n$ matrix, with $n$ on its diagonal, and $1$ on all other positions. What are the eigenvalues? How to find them?
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Your matrix commutes with the cyclic rotation operator of the $n$ elements. Therefore, it is diagonal in the Fourier basis. Use the Fourier modes to find the eigenvalues. – Zhuoran He Jan 12 '18 at 01:57
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See https://math.stackexchange.com/a/2178024/265466. This is essentially a duplicate of that question and many others. – amd Jan 12 '18 at 02:14
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Your matrix $A$ is equal to $(n-1)I+E$, where $E$ is the matrix with all its entries equal to $1$.
As $E^2=nE$, the only possible eigenvalues are $0$ and $n$. Both can easily be realized (that is, it is easy to find eigenvectors for both). In fact, $0$ has multiplicity $n-1$, and $n$ has multiplicity $1$.
In summary, the eigenvalues of $A$ are $n-1$ and $2n-1$.
Martin Argerami
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I understand why 0 and n are eigenvalues of E(E is singular and E also acts kind like projection matrix), but how to find their multiplicity? – zanyman Jan 12 '18 at 08:05
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