Let $G$ be a finite or compact group and $\rho: G \to \mathrm{U}(d)$ a $d$-dimensional unitary representation of $G$. If $\rho$ is irreducible then the following seems to be true:
$$
\mathrm{span}\_\mathbb{C} \{\rho(g) : g \in G\} = \mathbb{C}^{d \times d}
$$
i.e., the complex linear span of $\rho(g)$, taken over all $g \in G$, coincides with the set of all $d \times d$ complex matrices.


Is this claim true and does this result have a name?



Is the linear span of irrep matrices a complete matrix basis?
This is known as Burnside's theorem. Nowadays people formulate it as any algebra of matrices over an algebraically closed field acting irreduciblly is the whole matrix algebra. 


