Fundamental matrix

Definition

The fundamental matrix is a matrix that describes the expected number of times a Markov chain will be in a transient state before being absorbed in an absorbing state. It plays a critical role in analyzing absorption processes by providing insights into the behavior of transient states and their eventual transition to absorbing states.

5 Must Know Facts For Your Next Test

1. The fundamental matrix is denoted as $$N = (I - Q)^{-1}$$, where $$I$$ is the identity matrix and $$Q$$ is the transition matrix for transient states.
2. Each entry in the fundamental matrix represents the expected number of visits to a transient state before absorption occurs.
3. The fundamental matrix is only applicable in systems that include both transient and absorbing states.
4. The existence of the fundamental matrix implies that it is possible to analyze the long-term behavior of transient states in relation to absorbing states.
5. Using the fundamental matrix, one can compute various performance metrics, such as the expected time until absorption and the probability distribution over transient states.

Review Questions

• How does the fundamental matrix help in understanding the behavior of transient states in a Markov chain?
  • The fundamental matrix provides valuable information on how often transient states are visited before transitioning to an absorbing state. By calculating the expected number of visits to each transient state, we gain insights into their significance and duration within the overall process. This understanding allows for better predictions regarding system dynamics and helps identify which states may have a longer-lasting impact.
• What conditions must be met for a fundamental matrix to exist, and how does this impact analysis of Markov chains?
  • For a fundamental matrix to exist, there must be at least one absorbing state within the Markov chain, along with at least one transient state. If these conditions are met, it ensures that all transient states will eventually lead to absorption. This greatly impacts analysis, allowing us to compute important metrics such as expected visits and time until absorption, facilitating deeper insights into system behavior.
• Evaluate how changes in transition probabilities affect the entries of the fundamental matrix and what this means for system behavior.
  • Changes in transition probabilities can significantly alter the entries of the fundamental matrix. For instance, increasing transition probabilities towards absorbing states may reduce the expected number of visits to transient states, indicating quicker absorption. Conversely, if transitions away from absorbing states increase, it could lead to more prolonged interactions with transient states. This evaluation highlights the importance of understanding dynamics within Markov chains as these probabilities shape long-term outcomes.

Related terms

absorbing state: An absorbing state is a state in a Markov chain that, once entered, cannot be left. It signifies the end of the process.

transient state: A transient state is a state that can be left and may not be returned to. Over time, the probability of being in this state decreases.

expected number of steps: This refers to the average number of steps taken by the Markov chain to reach an absorbing state from a transient state.