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The eigenvalues and eigenvectors of $2^n$
$2^n$ is a real symmetric matrix with $3$ distinct eigenvalues, $1, 1^n, 2^{n-1}$ and $2^n$.
I need to show that the eigenvectors are a permutation of
$(-1, 1, 1, 1, 1, \ldots, 1)^\top$
and
$(1, 1, 1, 1, 1, \ldots, 1, 1, 1, 1, \ldots, 1)^\top$
I believe the first eigenvector should be $(-1, 1, 1, \ldots, 1)$ since it should satisfy the sum of the row of any eigenvalue is $0$. But how can I show that the other eigenvectors also satisfy this sum?
A:
We can use the characteristic polynomial method: the characteristic polynomial of a symmetric matrix is the product of its eigenvalues. Now there are $n$ zeroes in $\lambda – 1$, and $\lambda^n$ in $\lambda – 1$, so the trace of $\lambda I – A$ is $1 + n + \lambda^n – n\lambda$. Since $A$ has three distinct eigenvalues (and $n$ more to come), this is a cubic
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