Übertragungsmatrix

Übertragungsmatrix is a German term that translates to "transfer matrix" or "transition matrix" in English. It is a mathematical concept used in various fields to describe the relationship between states of a system over time or across stages. Essentially, it represents how a system's properties or values transition from one point to another.

In linear algebra, an Übertragungsmatrix is a square matrix that, when multiplied by a vector representing the current state of a system, yields a vector representing the next state. This is particularly useful for modeling discrete-time systems. For instance, in population dynamics, an Übertragungsmatrix can describe how the population of different age groups changes from one year to the next. Similarly, in electrical engineering, it can represent the transformation of voltage and current signals between different stages of a circuit.

The concept extends to areas like Markov chains, where the Übertragungsmatrix represents the probabilities of transitioning between different states. In this context, each element of the matrix indicates the likelihood of moving from one specific state to another. The sum of probabilities in each row of the matrix typically equals one, signifying that from any given state, there is a certainty of transitioning to some other state, including staying in the same state. The application of an Übertragungsmatrix allows for the prediction of future states or the analysis of long-term behavior of a system.