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Let V be a vector space of dimension 8 and f (endomorphism) such that the minimal polynomial of f is x^7. If B={v1,...,v8} is the Jordan basis of f, find the Jordan form and a Jordan basis for f^2 and f^3.

The attempt at a solution

Ok, I am having some trouble to solve this. My ideas are:

The minimal of f is x^7, then f is nilpotent of index 7. The only eigenvalue of the Jordan form is 0 and the multiplicity of 0 as a root of the minimal tells me the biggest size of the jordan blocks, in this case is a 7x7 block. As the Jordan matrix is an 8x8 matrix, then, it can only have a 7x7 block and a 1x1 block. Now, if the minimal of f is x^7, then the minimal of f^2 and of f^3 have to be x^4 and x^3 respectively. So, the biggest blocks in f^2 and f^3 are of size 4x4 and 3x3 respectively.

Now, I don't know how to get more information about the blocks of each Jordan form and I don't have a clue about the Jordan basis. I would appreciate any help. And sorry if I made any grammar or expression mistakes, english is not my native language. Thanks!