Let $T:\mathcal{P}(\mathbb{R})\to \mathcal{P}(\mathbb{R})$ such that $T(p)=p-p'$. Find all eigen values and eigen vectors of $T$.

Question

Let $T:\mathcal{P}(\mathbb{R})\to \mathcal{P}(\mathbb{R})$ such that $T(p)=p-p'$. Find all eigen values and eigen vectors of $T$.

Please someone help me with this problem. Thank you

Answer 1

I will assume $\mathcal{P}(\mathbb{R})$ is the set $\mathbb{R}[x]$ of polynomials with real coefficients. Then if $p$ is an eigenvector of $T$, we have $T p = p - p' = \lambda p$. By comparing the leading coefficients, we have that $\lambda =1$, so that $p'=0$, and $p=c$ for some constant $c$. Thus there is only one eigenvalue, $\lambda=1$, with corresponding eigenspace $\langle 1 \rangle$.