**How does an open quantum system effect the eigenvalues and energies? and how does this effect their matricies/vectors?**

An open quantum system is one that is distinct from of its environment but can also interact with its environment. The eigenvalues and eigenvectors of the open quantum system are defined in the same way for a closed quantum system. The only difference between the eigenbasis for an open quantum system and a closed quantum system is that the eigenbasis for a closed quantum system tells the complete story. An arbitrary state of the open quantum system can be written as a linear combination of its eigenstates (eigenvectors). However, if you want to understand the full behavior of the open quantum system with its environment, then you need, in principle, a basis of eigenstates that represents the combined system which now includes the environment. Often-times, that is not possible, so we need to account for interactions with the environment in other ways without full knowledge of the eigenstates that describe the whole combined system. How to do that is complicated and is an ongoing area of research, which is of particular importance for quantum information sciences.