linear algebra - 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?

Answer

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.