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A stochastic matrix describes a Markov chain X t over a finite state space S with cardinality α.. If the probability of moving from i to j in one time step is Pr(j|i) = P i,j, the stochastic matrix P is given by using P i,j as the i-th row and j-th column element, e.g.,
The state-transition matrix is used to find the solution to a general state-space representation of a linear system in the following form ˙ = () + (), =, where () are the states of the system, () is the input signal, () and () are matrix functions, and is the initial condition at .
As stated earlier, from the equation =, (if exists) the stationary (or steady state) distribution π is a left eigenvector of row stochastic matrix P. Then assuming that P is diagonalizable or equivalently that P has n linearly independent eigenvectors, speed of convergence is elaborated as follows.
The state space or phase space is the geometric space in which the axes are the state variables. The system state can be represented as a vector, the state vector. If the dynamical system is linear, time-invariant, and finite-dimensional, then the differential and algebraic equations may be written in matrix form.
Even with time-inhomogeneous Markov chains, where multiple transition matrices are used, if each such transition matrix exhibits detailed balance with the desired π distribution, this necessarily implies that π is a steady-state distribution of the Markov chain.
In probability theory, the matrix analytic method is a technique to compute the stationary probability distribution of a Markov chain which has a repeating structure (after some point) and a state space which grows unboundedly in no more than one dimension.
Given a linear time-invariant (LTI) system represented by a nonsingular matrix , the relative gain array (RGA) is defined as = = (). where is the elementwise Hadamard product of the two matrices, and the transpose operator (no conjugate) is necessary even for complex .
Steady state determination is an important topic, because many design specifications of electronic systems are given in terms of the steady-state characteristics. Periodic steady-state solution is also a prerequisite for small signal dynamic modeling. Steady-state analysis is therefore an indispensable component of the design process.