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# hw11 - has distinct eigenvalues-5 3 0 2 4 with...

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HW # 11 Due date: 5/10/2011 1. Suppose T ∈ L ( V ) is an operator and e 1 ,...,e n is an orthonormal basis of V. Prove that trace ( T ) = n X 1 h Te i ,e i i 2. Consider the operator T on the vectorspace of polynomials of degree 3 with complex coeﬃcients, P (3 , C ), given by f 7→ f 0 . Calculate trace ( T ). 3. Prove that the determinant of a nilpotent operator is 0. 4. Prove that the trace of a nilpotent operator is 0. (You may consider only the case of complex vectorspaces if you think that is easier). 5. Suppose T ∈ L ( C 11
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Unformatted text preview: ) has distinct eigenvalues -5, 3, 0, 2, 4 with multi-plicities 2, 2, 1, 3, 3. Suppose the eigenvalues -5, 3, 2 have 2 linearly independent eigenvectors, while the remaining eigenvalues 0, 4 have only one eigenvetor (up to scalar multiples). Find a possible Jordan form matrix of T. What is the characteristic polynomial of T? What is the determinant? Page 244: 8,11,12,13,14,16 Hint for 16: you may want to use number 1 above. 1...
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