These are two exercises from Hoffman/Kunze book Linear Algebra, chapter 6, section 3.
First: Let be an n-dimensional vector space and let be a linear operator on . Suppose that there exists some positive integer so that . Prove that .
Second: Let be an matrix with characteristic polynomial .
Show that .
These are my solutions, respectively:
If there exists some positive integer such that , then the minimal polynomial for has degree with . Since the minimal and characteristic polynomials have the same roots by the Cayley-Hamilton theorem, and that the characteristic polynomial has degree and also annihilates T, we have .
If has as characteristic polynomial then it is similar to a triangular matrix of the form where the blocks are size.
Thus, the trace of this matrix is which is the same trace as the original matrix, since if , then .
I'm sorry if I can't post two questions in the same thread, in that case please give me a heads up and I shall create a new one, also apologize for the length.
Is my reasoning correct?
Could I have worded it better?
Did I lack information?
In the second problem, is there a way could I do it without using the similarity to the triangular form (given that it's proved in the next chapter that if is an algebraically closed field then every matrix is similar to a triangular matrix)?
I'm currently a math bachelor and I had problems with this subject last semester, leading to a flunk (there were professor issues as well) and I know having a great grasp of this is essential for everything that follows, plus I'm really dedicated to do my best and understand this thoroughly.