Eigen values of matrix formed by column vector multiplied by row vector.

Question

Let $u$ and $v$ be column vectors in $\mathbb{R}^n$. Let $A = uv^T$ that is matrix formed by column vector multiplied by row vector. What are all the eigen values and eigen vectors of $A$? What is the rank of $A$?

Answer 1

If one of $u$ and $v$ is zero, then $A=0$ and the case is trivial.

Suppose then $u\ne0$ and $v\ne0$. Then $A\ne0$ and has rank at most $1$ (the rank of a product can't be greater than the rank of the factors). So the rank is $1$.

Consider $x=v$; then $uv^Tv=(v^Tv)u$ by direct computation.

The other eigenvalue is $0$. Since $uv^Tx=(v^Tx)u$, you should be able to finish.