HW12 Solutions MATH 110 Fall 2016-1 - HW 12 Solutions MATH...

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HW 12 Solutions, MATH 110 with Professor Stankova Find the Jordan normal form of the following matrices.
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HW 12 Solutions, MATH 110 with Professor Stankova λ = ω 3 , (1 , ω 3 , ω, ω 4 , ω 2 ) λ = ω 4 , (1 , ω 4 , ω 3 , ω 2 ) So the Jordan normal form is 1 0 0 0 0 0 ω 0 0 0 0 0 ω 2 0 0 0 0 0 ω 3 0 0 0 0 0 ω 4 Find a matrix Q such that Q - 1 AQ is a Jordan matrix where A = 2 - 1 2 2 2 1 - 1 2 2 . The characteris- tic polynomial is ( t - 3)( t 2 - 3 t +9). If ω is a third root of unity, then the roots of the quadratic factor are 3( ω +1) = - 3 ω 2 and 3( ω 2 +1) = - 3 ω (recall that 1+ ω + ω 2 = 0). The eigenvector for t = 3 is (1 , 1 , 1) t . For the quadratic factor, we can compute it directly or try to find a shortcut by playing around. Let’s try, just for kicks, to compute A (1 , ω, ω 2 ) t = - 3( ω, ω 2 , 1) t , and A (1 , ω 2 , ω ) t = - 3( ω 2 , ω, 1). Thus, we have Q = 1 1 1 1 ω ω 2 1 ω 2 ω Find the Jordan normal form of the matrix 0 1 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 1 1 - 1 - 2 2 1 The characteristic polynomial is t 5 - t 4 - 2 t 3 + 2 t 2 + t - 1 = ( t - 1) 3 ( t + 1) 2 . We have that A + I = 1 1 0 0 0 0 1 1 0 0 0 0 1 1 0 0 0 0 1 1 1 - 1 - 2 2 2 has rank 4. We know that ( A + I ) 2

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