Proof that the Trace of a Matrix is the sum of its Eigenvalues for non diagonalizable matrices

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Proof that the Trace of a Matrix is the sum of its Eigenvalues How does the proof given by Prof.Ted Shriffin for the above question work when the matrix is non diagonalizable? In the video which has the same proof https://www.youtube.com/watch?v=OLl_reBXY-g&list=PLlXfTHzgMRUIqYrutsFXCOmiqKUgOgGJ5&index=20 Prof.Pavel says that we have to deal the proof in a more subtle way when the matrix is non diagonalizable.