Show that there exists $Y \in \mathbb{C}^n$ such that $P$ is the characteristic polynomial of $A+XY^{T}$

Question

Let $A$ be a $n \times n$ matrix and $X$ a vector of $\mathbb{C}^n$ such that $(X,AX,\ldots,A^{n-1}X)$ is a basis of $\mathbb{C}^n$ and $P$ be a monic polynomial. Show that there exists $Y$ such that $P$ is the characteristic polynomial of $A + X\times Y^{t}$.

I did the case $n=1$, but I’m stuck further... Do you have some advices?

Answer 1

Since the effort shown in the original question is minimal, I will give rather some hints than a complete answer.

• Write out the matrix of $A$ under the basis $(X, AX, \dotsc, A^{n - 1}X)$. Also write out the coordinates of $X$ under this basis.

• Look at the companion matrix. How can you produce $P$ as a characteristic polynomial?

• Finally, how to undo the change of basis?