Depends on the type of a matrix...
In a functional matrix: The project manager and the functional manager share the project management work, where the functional manager will be responsible for the resource management part, and the project manager will be responsible for the rest. Roles and responsibilities must be clearly differentiated in order to avoid conflicts.
In a balanced matrix: There is power sharing between the project manager and the functional manager over the resources. In this case, resources will have two managers, which is a downside of this type of matrix (imagine a resource working on something for one manager, and then another manager telling him to stop and work on something else).
In a project matrix: The project manager has nearly full power over the resources when working on his project, and the functional manager's resource management responsibilities is limited to evaluations, advice, HR (approving vacations, but after they are already approved by the project manager), help with non-work related issues (such as getting a raise for this person).
Clearly, none of the above scenarios are satisfactory for both parties, that's why the relationship between a functional manager and a project manager is always suffering.