= Spectrum of an all-ones matrix
{title2=$\operatorname{spec}(J_n)=\{n,0\}\quad(n\geq2)$}
For $u=(1,\ldots,1)^T$, $J_n=uu^T$ maps $v$ to $(\sum_iv_i)u$. Therefore $u$ is an <eigenvector> with <eigenvalue> $n$, while the zero-sum <vector subspace> is its <kernel>. For $n\geq2$ a <basis> of the <kernel> is $e_1-e_n,\ldots,e_{n-1}-e_n$. These vectors and $u$ form an <eigenbasis>, proving <diagonalization> and <characteristic polynomial> $t^{n-1}(t-n)$. Both <algebraic multiplicity> and <geometric multiplicity> are one at $n$ and $n-1$ at zero. At $n=1$ the sole <eigenvalue> is 1; the zero-sum space has dimension zero and contributes no additional root.
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