Let T be a linear operator on a finite dimensional vector space V with Jordan co

Question

Let T be a linear operator on a finite dimensional vector space V with Jordan conical form

2 1 0 0 0 0 0
0 2 1 0 0 0 0
0 0 2 0 0 0 0
0 0 0 2 1 0 0
0 0 0 0 2 0 0
0 0 0 0 0 3 0
0 0 0 0 0 0 3

                     

(a) Find the characteristic polynomial of T.
(b) Find the dot diagram for each value of T.
(c) For each eigenvalues i, if any, does Ei = Ki?

(d) For each eigenvalue i find the smallest positive integer pi for which Ki = N((T-iI)pi)

(e) Compute the following numbers for each i, where Ui denotes the restriction of T-iI to Ki.

                (i) rank(Ui)

                (ii)rank(Ui2)

                (iii)nullity(Ui)

                (iv)nullity(Ui2)

Explanation / Answer

charasteristic polynomial =(x-2)^5*(x-3)^2 so eigen values are x=2 ,x=3

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