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Dynamical Systems and Markov Chains

Swati

New member
Oct 30, 2012
16
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
Jan 26, 2012
890
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
Oct 30, 2012
16
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
Jan 26, 2012
890
[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