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 Emeritus Sci Advisor PF Gold P: 16,101 Hrm. If a matrix A is diagonalizable, then I claim that there exists an n such that all of the nonzero eigenvalues of An lie in the right half-plane. The requirement that Tr(An) = 0, forces all the eigenvalues to be 0, and thus A is zero... clearly nilpotent! So the trick, then, is when the matrix is not diagonalizable. Then again, this only works for complex valued matrices.