This thesis deals with three methods of clustering a finite set of states of the Markov chain. All three methods are based on spectral analysis, i.e., on Perron-Frobenius cluster of eigenvalues of transition matrix. Each method is first theoretically analyzed and then tested on selected examples. As the best method, I would single out the GenPCCA method. First, it provides more useful infomation than other methods - the transition matrix of newly acquired Markov chain, the membership matrix which gives us clear information about how likely each state is to belong to a particular cluster. Furthermore, it seems to be the most accurate. In addition to the analysis of the spectrum and Schur vectors, the optimization procedure follows, which brings the solution even closer to the ideal one. The method that approaches to the Markov chain as a graph is also very precise in the analysis. It uses optimization methods in addition to the spectral analysis itself. However, the disadvantage is that it gives even the least information. The resulting membership matrix uniquely gives each state a unique cluster without information about other possible variants, and we do not get a new transition matrix or anything similar to that. The results mostly coincided with those obtained by the GenPCCA method. The method based on the analysis of singular values clustered the states very nicely in all three examples and the results coincided with those obtained by the GenPCCA method. It also gives us a coupling matrix that generally does not have a Markov property, but is a useful information. The method is probably intended to be the least precise as it uses only spectral analysis, without additional optimizations.