Tobias's answer is a good one from the standpoint of a PM trying to establish authority over a matrixed organization, which by its very nature is weak. I do not agree, however, with his summary statement:
In summary, the project manager can bind the team effectively only by
personal power via loyalty, respect, and so on.
I disagree with this statement not because it is incorrect but because an organization cannot solely rely on this to make the matrixed organization work. Developing the concept of personal power is extremely variable and an organization will most likely have PMs that are, at best, mediocre from this standpoint.
Instead, an organization needs to define the degree the degree of authority in a very formal way, documented, and agreed upon by the organization itself. In my limited experience, I find organizations do this poorly.
Documented Memorandum of Understanding that both the PM and the functional manager agree to and sign will establish worker expectations, time tables, degrees of utilization, what to do when there is a work crisis, how to resolve conflicts, how the employee is rewarded, how poor performance is handled, etc. It takes time to establish these constructs such that non compliance is easily identified and remedied. Doing this will establish the level of authority for both managers in a much better way and which neither can easily bypass.
EDIT BASED ON ADDITIONAL DETAIL:
Under a matrixed organization, an employee is tasked to both the functional and project managers in a pre-defined (if planned well) amount of time. This is what I meant by "degrees of utilization" with an established MOU. For example, an employee could be defined under the functional manager 75% and under the PM 25%. It could be defined also by the days of the week: M to W under the functional manager; Th and F under the PM.
Therefore, assigning tasks and priorities are not in conflict based on the agreed upon share.
If not defined, then your matrixed organization is broken.