Let U be a fixed n n matrix, and consider the operator T: Mnu M

Question:

Let U be a fixed n × n matrix, and consider the operator T: Mnu → M"" given by T(A) = UA.
(a) Show that A is an eigenvalue of T if and only if it is an eigenvalue of U.
(b) If A is an eigenvalue of T, show that EX(T) consists of all matrices w hose columns lie in Eλ(U):
Eλ(T) = ]P1, P2 - Pn][P, in EAU) for each i}
(c) Show that if dim[Eλ((7)] = d, then dim[Eλ(T)] = nd. [Hint: If B = {E1" Ed} is a basis of Eλ(U), consider the set of all matrices with one column from B and the other columns zero.