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annihilator annuls assume characteristic polynomial column commutes complementary completely reducible completes the proof complex numbers complex vector space define degree q denote dim U1 dim VT dimension dimensional direct sum eigenvalue eigenvector equations Euclidean re-space exists f(zu finite finite-dimensional follows Hence Hint ij-th entry induction integer irreducible irreducible polynomial Jordan Lecture 3-5 Let z1 linear combination linearly independent main diagonal mapping minimum polynomial monic polynomial nilpotent operator nonsingular nontrivial nonzero vector null space operator algebra ordered basis partial isometry permutation perpendicular projection Problems for Lecture projection with range proof of Theorem quotient space real numbers real vector space satisfying scalar multiple scalar operator self-adjoint set of vectors similar subset subspace invariant Suppose transpositions unitary space vectors in Euclidean z-th row zi+1 zn are linearly