Math Q&A Game

Hurkyl

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.

Hurkyl Recognitions: Gold Member Science Advisor Staff Emeritus	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.