# Thread: Normal orperator and eigenvalue

1. ## Normal orperator and eigenvalue

Let $V=R^2$ and define $T\in L(V)$ as the 90-degree rotation operator given by $T(x,y)=(-y,x)$ where $x,y \in R$ are the components of a vector in V.
a- show that T is normal
b-T does not have any eigenvalues in R.
Thanks

2. For a), find the matrix which represents $T$ with respect to the standard basis. Then show that the matrix is normal.

For b), suppose that $T(x,y)=\lambda(x,y)=(-y,x)$. What can you tell about $\lambda$? (Equivalently, find the roots of the characteristic polynomial of the matrix you found in part a) ).