- Thread starter
- #1

- Thread starter Swati
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- Thread starter
- #1

- Jan 26, 2012

- 890

Let \(P\) be an \(N\times N\) matrix, then \( N \rho \le 1\) so \(\rho \le 1/N\).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}.

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

CB

- Thread starter
- #3

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]

- Jan 26, 2012

- 890

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 \( \ge N\rho\)[FONT=MathJax_Math]how we get, N[/FONT][FONT=MathJax_Math]ρ[/FONT][FONT=MathJax_Main]≤[/FONT][FONT=MathJax_Main]1 [/FONT]

CB