Assume that $E$ is a vector bundle over a manifold $M$ with a connection $\nabla$. Is there a (unique) connection $\nabla'$ on $E' := Hom (E,E)$ with the following property: for every curve $\gamma$ which connects point $x$ to $y$, with $\nabla$ parallel transport $\phi$ and $\nabla'$ parallel transport $\psi$, we have $\psi(T)=\phi^{-1}T\phi$. Moreover if $\nabla$ is a Riemannian connection corresponding to a Riemannian metric on $E$, can we choose a Riemannian comnnection $\nabla '$ as above. In the latter we consider the Riemannian metric $tr(AB^{*})$ on the $E'$ bundle.