2.II.15G

(a) A complex $n \times n$ matrix is said to be unipotent if $U-I$ is nilpotent, where $I$ is the identity matrix. Show that $U$ is unipotent if and only if 1 is the only eigenvalue of $U$.

(b) Let $T$ be an invertible complex matrix. By considering the Jordan normal form of $T$ show that there exists an invertible matrix $P$ such that

$P T P^{-1}=D_{0}+N$

where $D_{0}$ is an invertible diagonal matrix, $N$ is an upper triangular matrix with zeros in the diagonal and $D_{0} N=N D_{0}$.

(c) Set $D=P^{-1} D_{0} P$ and show that $U=D^{-1} T$ is unipotent.

(d) Conclude that any invertible matrix $T$ can be written as $T=D U$ where $D$ is diagonalisable, $U$ is unipotent and $D U=U D$.

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