Vector bundle endomorphism ($A\in \Gamma (\mathrm{End}E)$)

A smooth vector bundle morphism from a vector bundle $E$ to itself, covering the identity on the base, is a vector bundle endomorphism. It is fibrewise linear and is equivalently a smooth section of $\mathrm{End}E$. In a local frame its matrix is smooth and changes by conjugation under changes of frame. An almost complex structure is a real endomorphism of the tangent bundle satisfying $J^2=-I$.