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Unformatted text preview: Homework assignment 11 Section 7.3 pp. 249250 Exercise 1. Let N 1 and N 2 be 3 3 nilpotent matrices over the field F . Prove that N 1 and N 2 are similar if and only if they have the same minimal polynomial. Solution: If N 1 and N 2 are similar, they have the same minimal polynomial (cf. pg. 192). Conversely, suppose that N 1 and N 2 have the same minimal polynomial. The minimal polynomial must be x k for some 1 k 3 . If k = 1 we get that the matrix is the zero matrix, so both N 1 and N 2 are the zero matrix. If k = 2 we get that the Jordan form for N 1 and N 2 is 1 and thus, they are similar. If k = 3 then the Jordan form of both matrices is 1 1 and thus, they are similar. Exercise 3. If A is a complex 5 5 matrix with characteristic polynomial f = ( x 2) 3 ( x + 7) 2 and minimal polynomial p = ( x 2) 2 ( x + 7) , what is the Jordan form for A ? Solution: The block matrix associated to the characteristic value 2 is a 3 3 matrix with 2 s along the diagonal with an elementary Jordan matrix of size 2 2 (the multiplicity of 2 in the minimal polynomial) as the first block. i.e.: 2 1 2 2 Analogously, the block matrix associated to the characteristic value 7 is a 2 2 matrix with 7 s along the diagonal with an elementary Jordan matrix of size 1 1 as the first block, i.e.: 7 7 Hence the Jordan form for A is: 2 1 2 2 7 7 1 Exercise 4. How many possible Jordan forms are there for a 6 6 complex matrix with characteristic polynomial ( x + 2) 4 ( x 1) 2 ? Solution: ATTENTION!!! There are 8 possibilities for the minimal polynomial, this implies that there are at least 8 different Jordan forms. But the minimal polynomial ( x + 2) 2 ( x 1) may correspond to TWO different matrices, namely  2 1 2 2 1 2 1 1 and  2 1 2 2 2 1 1...
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 Spring '10
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