Let $A\in GL_n(\mathbb{C})$. Show that $A$ has a unitary triangularization, that is there exists $U\in M(n\times n)$ unitary such that $U^{-1}AU$ is an upper triangular matrix.
Since the characteristic polynomial of $A$ splits over $\mathbb{C}$, I immediately know that $A$ is similar to an upper triangular matrix. I'm now trying to show that the according change of basis matrix is unitary. One approach was trying to show the sufficient condition that $\left|\left| Ux \right|\right|=\left|\left|x \right|\right|,\, \forall x\in\mathbb{C}^n$, which I haven't been able to show so far.
You can prove it by induction. If $n=1$, it is trivial. Let $v$ be an eigenvector of $A$. Now let $V_1=\Bbb Cv$ and let $V_2=V_1^\perp$. If $\{v_1,\ldots,v_{n-1}\}$ is an orthonormal basis of $V_1^\perp$, then $\{v/\|v\|,v_1,\ldots,v_{n-1}\}$ is an orthonormal basis of $\Bbb C^n$ and the matrix of $A$ with respect to that basis is of the form $\left[\begin{smallmatrix}\lambda&A\\0&B\end{smallmatrix}\right]$, where $\lambda\in\Bbb C$, and $B$ is a $(n-1)\times(n-1)$ matrix. Now, consider the map $A_1\colon V_2\longrightarrow V_2$ defined by $A_1=\pi\circ A$, where $\pi$ is the orthogonal projection from $V$ onto $V_2$. And now you apply the induction hypothesis to $A_1$.