2011-42.md

course	course_year	question_number	tags	title	year
Linear Algebra	IB	42	IB 2011 Linear Algebra	Paper 4, Section II, G	2011

Let $V$ be an $n$-dimensional $\mathbb{R}$-vector space and $f, g: V \rightarrow V$ linear transformations. Suppose $f$ is invertible and diagonalisable, and $f \circ g=t \cdot(g \circ f)$ for some real number $t>1$.

(i) Show that $g$ is nilpotent, i.e. some positive power of $g$ is 0 .

(ii) Suppose that there is a non-zero vector $v \in V$ with $f(v)=v$ and $g^{n-1}(v) \neq 0$. Determine the diagonal form of $f$.