Dynamical Systems and Markov Chains

Swati

New member
Prove that if $$P$$ is a stochastic matrix whose entries are all greater than or equal to $$\rho$$, then the entries of $$P^{2}$$ are greater than or equal to $$\rho$$.

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CaptainBlack

Well-known member
Prove that if P is a stochastic matrix whose entries are all greater than or equal to /{/rho}, then the entries of /{/P^{2}} are greater than or equal to /{/rho}.
Let $$P$$ be an $$N\times N$$ matrix, then $$N \rho \le 1$$ so $$\rho \le 1/N$$.

Now every element of $$P^2$$ is $$\le N \rho^2 \le \rho$$ etc

CB

Swati

New member
Let $$P$$ be an $$N\times N$$ matrix, then $$N \rho \le 1$$ so $$\rho \le 1/N$$.

Now every element of $$P^2$$ is $$\le N \rho^2 \le \rho$$ etc

CB

[FONT=MathJax_Math]how we get, N[/FONT][FONT=MathJax_Math]ρ[/FONT][FONT=MathJax_Main]≤[/FONT][FONT=MathJax_Main]1 [/FONT]

CaptainBlack

Well-known member
[FONT=MathJax_Math]how we get, N[/FONT][FONT=MathJax_Math]ρ[/FONT][FONT=MathJax_Main]≤[/FONT][FONT=MathJax_Main]1 [/FONT]
Depending on how the stochastic matrix is defined either the row or column sums are 1, but if every element is $$\ge \rho$$ then a row (column) sum $$\ge N\rho$$

CB