Assume that $E$ is a vector bundle over a manifold  $M$  with a connection $\nabla$.Is there a (unique) connection
$\nabla'$ on $Hom (E,E)$ 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 home bundle