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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?

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    $\begingroup$ Certainly there are finite-dimensional unitary representations of infinite groups in general. $\endgroup$ – whacka Oct 4 '15 at 23:27
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Every unitary matrix is diagonalizable, but $\rho(x)$ is not diagonalizable unless $x=0$ (its only eigenvector (up to scaling) is $(1,0)$).

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