Dynamical Systems and Markov Chains

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SUMMARY

This discussion focuses on the properties of stochastic matrices, specifically proving that if \(P\) is a stochastic matrix with all entries greater than or equal to \(\rho\), then the entries of \(P^{2}\) also meet this condition. It is established that for an \(N \times N\) matrix \(P\), the condition \(N \rho \le 1\) leads to \(\rho \le 1/N\). Consequently, it is shown that every element of \(P^2\) is bounded by \(N \rho^2\), confirming that \(P^2\) retains the property of having entries greater than or equal to \(\rho\).

PREREQUISITES
  • Understanding of stochastic matrices and their properties
  • Familiarity with matrix multiplication and its implications
  • Knowledge of basic linear algebra concepts
  • Comprehension of inequalities and their applications in matrix theory
NEXT STEPS
  • Study the properties of stochastic matrices in detail
  • Learn about matrix powers and their significance in Markov chains
  • Explore the implications of row and column sums in stochastic matrices
  • Investigate applications of Markov chains in various fields such as economics and computer science
USEFUL FOR

Mathematicians, data scientists, and anyone involved in the study of dynamical systems or Markov chains will benefit from this discussion, particularly those looking to deepen their understanding of stochastic processes.

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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