Let be matrices, real or complex. Define the trace to be the sum of diagonal entries . Define the commutator to be the difference . Give the definition of the eigenvalues of a matrix and prove that it can have at most two distinct eigenvalues. Prove that a) , b) equals the sum of the eigenvalues of , c) if all eigenvalues of are equal to 0 then , d) either is a diagonalisable matrix or the square , e) where and is the unit matrix.
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