  Markov matrices have an important role in the filed of stochastic processes.
In this paper, we will show and prove a series of conclusions on Markov
matrices and transformations rather than pay attention to stochastic processes
although these conclusions are useful for studying stochastic processes. These
conclusions we come to, which will make us have a deeper understanding of
Markov matrices and transformations, refer to eigenvalues, eigenvectors and the
structure of invariant subspaces. At the same time, we account for the
corresponding significances of the conclusions. For any Markov matrix and the
corresponding transformation, we decompose the space as a direct sum of an
eigenvector and an invariant subspace. Enlightened by this, we achieve two
theorems about Markov matrices and transformations inspired by which we
conclude that Markov transformations may be a defective matrix--in other words,
may be a nondiagonalizable one. Specifically, we construct a nondiagonalizable
Markov matrix to exhibit our train of thought.
