Can there be two matrices that have the same eigenvalues and corresponding eigenvectors? Lets say there is a matrix with eigenvalues 1 and 2 and eigenvectors $(1,2)^T$ and $(3,4)^T$, will there be another matrix with these? I want to intuitively say no, but I'm not positive.
It is possible for multiple matrices to have the same eigenvalues and eigenvectors as long as the diagonals have the same entries, but in a different order.
If the elements of the trace are equal, and the traces are equivalent, then the eigenvalues and eigenvectors will be the same.