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# solution10 - Homework assignment 10 Section 6.3 pp 197-198...

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Let V be a finite dimensional vector space. What is the minimal polynomial for the identity operator on V ? What is the minimal polynomial for the zero operator? The minimal polynomial for the identity operator is p ( x ) = x - 1 . It is monic, of degree 1 and it annihilates the identity operator. The minimal polynomial for the zero operator is p ( x ) = x . Let A be the 4 × 4 real matrix A = 1 1 0 0 - 1 - 1 0 0 - 2 - 2 2 1 1 1 - 1 0 Show that the characteristic polynomial for A is x 2 ( x - 1) 2 and that it is also the minimal polynomial. Calculating det( xI - A ) we get x 2 ( x - 1) 2 . The minimal polynomial should have the same degree 1 factors, i.e. x and ( x - 1) . Calculating the remain- ing possibilities we get: A ( A - I ) = 0 , A 2 ( A - I ) = 0 , A ( A - 1) 2 = 0 . Therefore the minimal polynomial is the characteristic polynomial. Is the matrix A of Exercise 3 similar over the field of complex numbers to a diagonal matrix? One can easily check that the matrices A and A - I have rank 3 . Hence, A has exactly two eigenvectors: one with eigenvalue 0 , and the other with eigenvalue 1 . So A does not have a basis of eigenvectors, and thus is not similar to a diagonal matrix over the complex field.

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solution10 - Homework assignment 10 Section 6.3 pp 197-198...

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