eigenvalues and eigenvectors of $vv^T$

Given a column vector $v$ in $\mathbb{R}^n$, what are the eigenvalues of matrix $vv^T$ and associated eigenvectors?

PS: not homework even though it may look like so.

Solution 1:

Assume $v \neq 0$. Then $v$ is an eigenvector with eigenvalue $|v|^2 >0$, since $(vv^t)v=v(v^t v)=v |v|^2 = |v|^2 v$, and any nonzero vector $x$ in the orthogonal complement of $v$ (which is of dimension $n-1$) is an eigenvector with eigenvalue zero, since $(vv^t)x = v(v^t x) = v(v \cdot x)=v0=\mathbf{0}=0x$.

Solution 2:

The columns of the matrix are $v_1v,\ldots,v_nv$ so if we take two column these one are linearly dependent, and so $vv^T$ has a rank of at most $1$. It's $0$ if $v=0$, and if $v\neq 0$, we have $\mathrm{Tr}A=|v|^2$ so the eignevalues are $0$ with multiplicity $n-1$ and $|v|^2$ with multiplicity $1$.