Question

Hello.

This is an exercise is from Hoffman, Linear Algebra, chapter 7.4; but it has no solution, Can you help me to understand how to solve it? I have just a very general idea of how to solve it and I am afraid that, if the degree of the polynomial is changed, I may fail the solution.

4. Construct a linear operator T with minimal polynomial x^2 (x - 1)^2 and characteristic
polynomial x^3(x-1)^4. Describe the primary decomposition of the vector
space under T and find the projections on the primary components. Find a basis
in which the matrix of T is in Jordan form. Also find an explicit direct sum decomposition
of the space into T-cyclic subspaces as in Theorem 3 and give the invariant
factors.

Best Regards.

Answer 1

Since we have to construct and in fact the Vector space on which is defined is not given. Hence we can simply consider (we choose dimension because given Characteristic polynomial is of degree ) & we can choose matrix of with respect to standard basis of to be a Jordan normal form. Further by given Characteristic polynomial we have are only eigenvalues of . So by theory of Jordan normal form we known that, size of the largest Jordan block corresponding to eigenvalue is (since the power of factor corresponding to in minimal polynomial has power ) and sum of sizes of Jordan blocks corresponding to eigenvalue is and similarly for eigenvalue , the size of largest Jordan block is and sum of sizes of block is . Further for eigenvalue we can choose remaining blocks of size . Hence we have obtained simplest JC form(matrix representation of )from which we can easily find out definition of . Hence our first question of construction of is completed.

Now by Primary decomposition theorem, the Vector space is direct sum of eigenspaces of . Since we know the matrix representation of   (which is JC form) we can easily find out eigenspaces of . Hence the Primary decomposition of Vector space is into generalized eigenspaces.

Now the corresponding projections are obvious because we are projecting on eigenspaces.

The basis in which is in Jordan form is the standard basis of as we had already mentioned earlier.  

(Further as you didn't written the which theorem 3 in question I can just give hint of last part) the last part follows from "the cyclic subspaces of are the subspaces corresponding to Jordan blocks.