Finite dimensional representation of infinite group cannot be unitary: example with $\mathbb R$

Question

Consider the representation of the group of real numbers $\mathbb R$ given by $$ \rho (x) = \begin{pmatrix} 1 & x \\ 0 & 1 \end{pmatrix} $$ for $x \in \mathbb R$. How can we see that this representation cannot be equivalent to a unitary representation?

Answer 1

Every unitary matrix is diagonalizable, but $\rho(x)$ is not diagonalizable unless $x=0$ (its only eigenvector (up to scaling) is $(1,0)$).