I had an interesting problem in class today. Find
I have a very strange question that I know is true, but seems to be bugging me.
Essentially, my first idea was to assume the opposite, i.e. that there is a polynomial such that p(A)=B, for example (I'd do the p(B)=A one separately).
Still, something makes me think there's a much more elegant solution that I'm not seeing.
I've encountered a problem while studying matrices.
Prove that there does not exist a polynomial with real coefficients such that p(A)=B or p(B)=A.
I've read up on eigenvalues, eigenvectors, characteristic polynomials and diagonalization, but nothing seems to be making sense, as the whole thing gets way too complicated for a high school problem.
I think there's a way to do this without using any of the aforementioned.
Can you help me out, please?