Dynamical Systems and Markov Chains

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Discussion Overview

The discussion revolves around properties of stochastic matrices, specifically examining the implications of having all entries greater than or equal to a certain threshold, \(\rho\). Participants are exploring the relationship between the entries of a stochastic matrix \(P\) and its square \(P^2\), focusing on whether the entries of \(P^2\) maintain the same lower bound as those of \(P\). The scope includes mathematical reasoning and proofs related to dynamical systems and Markov chains.

Discussion Character

  • Mathematical reasoning
  • Debate/contested

Main Points Raised

  • One participant asserts that if \(P\) is a stochastic matrix with entries \(\ge \rho\), then the entries of \(P^2\) should also be \(\ge \rho\).
  • Another participant provides a condition that \(N \rho \le 1\) implies \(\rho \le 1/N\), suggesting that this condition is necessary for the argument.
  • There is a claim that every element of \(P^2\) is \(\le N \rho^2\), which raises questions about the validity of the initial assertion regarding the lower bound.
  • Discussion includes the clarification that depending on the definition of the stochastic matrix, either the row or column sums equal 1, which leads to implications for the sums of the entries being \(\ge N\rho\).

Areas of Agreement / Disagreement

Participants express differing views on the implications of the conditions set for the stochastic matrix and the resulting properties of \(P^2\). The discussion remains unresolved, with no consensus reached on the validity of the claims made.

Contextual Notes

There are limitations regarding the assumptions made about the stochastic matrix and the definitions used, particularly concerning row and column sums. The mathematical steps leading to conclusions about the entries of \(P^2\) are not fully resolved.

Swati
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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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Swati said:
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
 
CaptainBlack said:
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=MathJax_Math]ρ[FONT=MathJax_Main]≤[FONT=MathJax_Main]1
 
Swati said:
[FONT=MathJax_Math]how we get, N[FONT=MathJax_Math]ρ[FONT=MathJax_Main]≤[FONT=MathJax_Main]1

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
 

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