A square matrix (of some size ) satisfies the condition
(a) Show that this matrix is similar to a diagonal matrix.
(b) Show that for every positive integer there exists a matrix satisfying the above condition with .
I have little or no idea where to go with this problem. I know that for (a), is obviously a diagonal matrix , but am not sure if this is of any use. I also am not sure if I can simply say , so , in which case, either (i) or , or (ii) with and , and in this case, I'm not sure how to use (ii) to prove anything. Obviously (b) has something to do with the second term being -8A, but apart from this I can't really think of anything.
Any help would be greatly appreciated.
Thanks. I have no idea, however, what an annihilator polynomial is, and hadn't come across the term before, and so I don't think this was a method which my lecturer intended me to use or would have accepted without proofs (this was an exam question in a paper I had a few months ago).