Linear Mathematics | Part IB, 2002

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

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

PTP1=D0+NP T P^{-1}=D_{0}+N

where D0D_{0} is an invertible diagonal matrix, NN is an upper triangular matrix with zeros in the diagonal and D0N=ND0D_{0} N=N D_{0}.

(c) Set D=P1D0PD=P^{-1} D_{0} P and show that U=D1TU=D^{-1} T is unipotent.

(d) Conclude that any invertible matrix TT can be written as T=DUT=D U where DD is diagonalisable, UU is unipotent and DU=UDD U=U D.

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