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Methods from Markov systems

All methods from Markov systems.

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Evolve a Markov system by diagonalisation

For

x(k+1)=Ax(k),\mathbf{x}(k+1)=A\mathbf{x}(k),

repeated substitution gives

x(k)=Akx(0).\mathbf{x}(k)=A^k\mathbf{x}(0).

If A=PDP−1A=PDP^{-1}, compute this as

x(k)=PDkP−1x(0).\mathbf{x}(k)=PD^kP^{-1}\mathbf{x}(0).

In an eigenbasis, the eigenvalue-11 component is unchanged, while every mode with modulus less than 11 decays, as in stability of systems. This is the calculation from matrix powers with the state vector read as probabilities.

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Evolve a Markov system by diagonalisation