A **transfer matrix** is a mathematical tool used in various fields, notably in physics, to analyze a system or process by relating the state of a system at one point to its state at another point. The concept is widely applied in statistical mechanics, condensed matter physics, quantum mechanics, and in the field of linear systems.
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A transfer matrix stores the weights of allowed transitions between finitely many boundary states. Its powers sum weights of compatible sequences of transitions. For a cyclic boundary condition, the matrix trace counts weighted closed sequences of length .