A transition matrix is a mathematical tool that represents a set of probabilities of moving from one state to another in a finite set of states. It is also called a stochastic matrix, a probability matrix, or a magic matrix. The term stochastic refers to the random nature of the transitions, while probability and magic refer to the likelihood of a particular transition occurring. Other synonyms for transition matrix include Markov matrix, chain matrix, state matrix, and adjacency matrix. In summary, a transition matrix is a versatile tool that has various applications in fields such as statistics, economics, engineering, and computer science, among others.