Prove that $T$ and $L\lambda$ satisfy...

Question

$\mathsf{Lemma\;34.3}$: Let $T$ be a linear operator on a finite dimensional complex vector space.

Answer 1

Let $n$ be a non-negative integer. Then $$ TL_\lambda^n = T\sum_{i=0}^n \binom{n}{i}T^i (\lambda I)^{n-i} = \sum_{i=0}^n \binom{n}{i}T^{i+1} (\lambda I)^{n-i} = \left(\sum_{i=0}^n \binom{n}{i}T^i (\lambda I)^{n-i}\right)T = L_\lambda^n $$ since all matrices commute with powers of themselves and with the identity. In particular, the equality holds for $n=1$, and so the first two properties you are to show hold.

Now, \begin{align} L_\lambda L_\mu &= (T - \lambda I)(T - \mu I) \\ &= T^2 - \mu T - \lambda T + \lambda \mu I \\ &= T^2 - \lambda T -\mu T + \mu \lambda I \\ &= (T - \mu I)(T - \lambda I) \\ &= L_\mu L_\lambda \end{align} establishes the third property.