Diagonalizable Proof Homework: True or False?

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SUMMARY

The discussion centers on the diagonalizability of linear operators T and U defined on the space of n x n matrices over a field F, given a fixed n x n matrix A. It is established that if A is diagonalizable, then T(B) = AB is also diagonalizable. However, the claim that U(B) = AB - BA is diagonalizable when A is diagonalizable is false. The reasoning involves analyzing the effects of T and U on basis matrices when A is diagonal.

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Homework Statement



Let V be the space of n X n matrices over F. Let A be a fixed n X n matrix
over F. Let T and U be the linear operators on V defined by
T(B) = AB
U(B) = AB - BA.
1. True or false? If A is diagonalizable (over F), then T is diagonalizable.
2. True or false? If A is diagonalizable, then U is diagonalizable
Thanks for the help.



The Attempt at a Solution


I'm guessing that 1 is true and 2 is false. I'm not sure, since these are linear operators rather than simple matrices.
 
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Consider, first, the case in which A is diagonal. What do T and U do to the "basis" matrices?
 

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