MA554_AUG09

MA554_AUG09 - Math 554 Qualifying Exam August 2009(Jiu-Kang...

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Math 554 Qualifying Exam. August, 2009 (Jiu-Kang Yu) 1. Let J e be the e × e complex matrix with J j +1 ,j = 1 for j = 1 ,...,e - 1, J i,j = 0 if i 6 = j + 1. It is the so-called e × e nilpotent Jordan block. Let e 1 ≥ ··· ≥ e r be a decreasing sequence of positive integers and let A = J e 1 ,...,e r be the direct sum of J e 1 ,...,J e r . (a) (8 points) Compute dim ker A m , for m 0. (b) (8 points) Show, without using the structure theorem, that if J e 1 ,...,e r is similar to J f 1 ,...,f s (where f 1 ≥ ··· ≥ f s is another decreasing sequence of positive integers), then r = s and e i = f i for i = 1 ,...,r . (c) (8 points) What is the Jordan form of A 2 ? It is enough to describe the sizes (and eigenvalues) of its Jordan blocks. (d) (8 points) What is the Jordan form of A 2 + A ? 2. (10 points) Let F 2 be the finite field Z / 2 Z . How many similarity classes of 3 × 3 invertible matrices over F 2 are there? You may use the fact that there are 2,1,2 monic irreducible polynomial
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