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(Solved): (40 points) (a) Let T:V->V be a linear operator such that TT^(*)=\sigma Id for a positive real nu ...
(40 points)
(a) Let T:V->V be a linear operator such that TT^(*)=\sigma Id for a positive real number \sigma . Show
that T is normal.
(b) Consider the linear operator T:C^(3)->C^(3) with the standard inner product, defined by
T(x)=Ax, where
A=[[4+3i,4i,-6-2i],[-4i,4-3i,-2-6i],[6+2i,-2-6i,1]]
Show that T is normal and find an orthonormal eigenbasis for T.