Q: What does it imply if all the eigenvalues of a matrix are all the same? What properties does a matrix have if all its eigenvalues are the same? In particular, what happens if all eigenvalues are all equal to 1?

A: Generally, it means not much in paricular, just that it is composed of Jordan block corresponding to the same eigenvalue. From the Jordan decomposition theorem, we see that $A = V^{-1} J V$, with $J$ having constant diagonal entries, which are the eigenvalues of $A$.
However, if the matrix has all the eigenvalues the same, and is in addition normal, you know that it is a constant multiple of identity matrix.
