I'm having trouble with this question, I'd like someone to point me in the right direction.
let $A$ be a n by n matrix with real values. show that there is another n by n real matrix $B$ such that $B^3=A$, and that $B$ is symmetric. Are there more matrices like this $B$ or is it the only one?
What I was thinking:
I don't have a clear way to solve it. I think we need to use the fact that if a real matrix is symmetric, then it is normal, and so has an orthonormal basis of eigenvectors...Other then that I don't really know anything.