Characteristic polynomial of the matrix $(a_i\cdot b_j)$ Let $a=(a_1,\ldots,a_n)$ and $b=(b_1,\ldots,b_n)$ be two real vectors.
Then the characteristic polynomial of the matrix $(a_i\cdot b_j)_{i,j=1\ldots n}$ is 
$$\chi(x) = x^{n}- \left<a,b\right>\cdot{}x^{n-1}.$$
Does anyone have a short proof of this result? Or, even better, a reference to it? I have a proof, but I find it clumsy.

Denote your matrix by $A$. Then: 


*

*Show that any vector orthogonal to $b$ is an eigenvector of $A$ with eigenvalue equal to $0$. How many such linearly independent vectors do we have?

*Show that $a$ is also an eigenvector of $A$. What is the corresponding eigenvalue?

*Combine previous two results. 
