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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}.
[FONT=MathJax_Math]how we get, N[/FONT][FONT=MathJax_Math]ρ[/FONT][FONT=MathJax_Main]≤[/FONT][FONT=MathJax_Main]1 [/FONT]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
Swati said:[FONT=MathJax_Math]how we get, N[/FONT][FONT=MathJax_Math]ρ[/FONT][FONT=MathJax_Main]≤[/FONT][FONT=MathJax_Main]1 [/FONT]
Dynamical systems are mathematical models used to describe the behavior of a complex system over time. They involve a set of equations that define how the system changes or evolves over time based on its current state.
Discrete dynamical systems are characterized by a finite or countably infinite set of states and a discrete time variable, meaning that the system only changes at specific time intervals. Continuous dynamical systems, on the other hand, have an infinite set of states and a continuous time variable, meaning that the system changes continuously over time.
A Markov chain is a type of dynamical system that models a sequence of events where the probability of each event only depends on the previous event. In other words, the future state of the system is only dependent on the current state, not the entire history of the system.
Markov chains are used in a variety of fields, including finance, biology, and computer science. They are commonly used to model and predict the behavior of complex systems, such as stock prices, genetic mutations, and internet traffic patterns.
One limitation of dynamical systems is that they are only as accurate as the assumptions and equations used to model the system. If these assumptions are incorrect or incomplete, the model will not accurately reflect the behavior of the system. Additionally, Markov chains are limited by the assumption that the future state of the system is only dependent on the current state, which may not always hold true in real-world scenarios.