Let $L/K$ be a Galois extension and $V$ be a finite dimensional vector space over the field $K.$ Then $V \otimes_K L $ is a $L$ vector space. If $T : V \to V$ be any $K$-linear map then $T\otimes Id : V \otimes_K L \to V \otimes_K L $ is $L$-linear map. Then I want to know if there is any connection between the eigen values of $T$ in $L$ and the eigen values of $T \otimes Id.$
Is there any reference to know about that. Thank you.