2.II.15G
(a) A complex matrix is said to be unipotent if is nilpotent, where is the identity matrix. Show that is unipotent if and only if 1 is the only eigenvalue of .
(b) Let be an invertible complex matrix. By considering the Jordan normal form of show that there exists an invertible matrix such that
where is an invertible diagonal matrix, is an upper triangular matrix with zeros in the diagonal and .
(c) Set and show that is unipotent.
(d) Conclude that any invertible matrix can be written as where is diagonalisable, is unipotent and .
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