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

 

[blue_raver22]

Dynamical systems and Markov chains are important concepts in the field of mathematics and physics. A dynamical system is a mathematical model that describes the time evolution of a physical system, while a Markov chain is a mathematical model that describes a sequence of events where the probability of each event depends only on the previous event.

The statement 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 \(\rho\) is known as the Perron-Frobenius theorem. This theorem is a fundamental result in the theory of dynamical systems and has been extensively studied and proven by mathematicians.

To prove this statement, we first need to understand the properties of stochastic matrices. A stochastic matrix is a square matrix whose entries are all non-negative and each row sums to one. This means that the entries represent probabilities of transitioning from one state to another in a Markov chain.

Now, let us consider the matrix \(P^{2}\). The entry in the \(i^{th}\) row and \(j^{th}\) column of \(P^{2}\) represents the probability of transitioning from state \(i\) to state \(j\) in two steps. This can be written as the sum of probabilities of transitioning from state \(i\) to any intermediate state \(k\), and then from state \(k\) to state \(j\). Mathematically, this can be represented as:

\((P^{2})_{ij} = \sum_{k=1}^{n} P_{ik}P_{kj}\)

Since \(P\) is a stochastic matrix, each entry \(P_{ik}\) and \(P_{kj}\) is greater than or equal to \(\rho\). Therefore, the sum of these entries will also be greater than or equal to \(\rho\). This implies that \((P^{2})_{ij} \geq \rho\).

Hence, we can conclude that if \(P\) is a stochastic matrix whose entries are all greater than or equal to \(\rho\), then the entries of \(P^{2}\) are also greater than or equal to \(\rho\). This result is in accordance with the Perron-Frobenius theorem and further strengthens the understanding of dynamical systems and Markov chains.