Exercise 20

Suppose that $T$ is $n\!\times\!n$ and $c,d$ are scalars.

• Prove that if $T$ has the eigenvalue $\lambda$ with an associated eigenvector $\vec{v}$ then $\vec{v}$ is an eigenvector of $cT+dI$ associated with eigenvalue $c\lambda+d$.

• Prove that if $T$ is diagonalizable then so is $cT+dI$.