Dynamical Systems and Markov Chains

[Swati]

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 \(\rho\).

 

[CaptainBlack]

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]

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]

[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