Overgangsmatrixen

Overgangsmatrixen, also known as a transition matrix, is a fundamental concept in linear algebra and discrete dynamical systems. It is a square matrix where each entry represents the probability or rate of transition from one state to another within a system. The rows typically represent the initial states, and the columns represent the subsequent states. The sum of probabilities in each row must equal one, indicating that from any given state, the system must transition to some state.

The primary application of a transition matrix is to model the evolution of a system over time. If a vector represents the probability distribution of states at a certain time, multiplying this vector by the transition matrix yields a new vector representing the probability distribution of states at the next time step. Repeated multiplication of the initial state vector by the transition matrix allows for the prediction of the system's long-term behavior, including its steady state or equilibrium distribution if one exists.

Transition matrices are widely used in various fields. In population dynamics, they model how populations of different species change over generations. In economics, they can describe the flow of capital between different sectors. In computer science, they are employed in algorithms like Google's PageRank to determine the importance of web pages. The properties of the transition matrix, such as its eigenvalues and eigenvectors, provide crucial insights into the dynamics of the system it represents.