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A transition matrix is a mathematical tool used to describe changes from one state to another in a system. It is often represented as a square matrix where each entry indicates the probability of transitioning from one state to another. Typically in our context, each column of the matrix adds up to 1, representing total probability distribution among possible transitions.
A transition matrix is essential when dealing with Markov chains, where future states depend only on the current state and not past states. Using this matrix, we can determine long-term behavior of a system by analyzing its effect on probability distributions of states, like finding the steady-state vector, which remains constant as time progresses.
For instance, consider a transition matrix:

• The element at the first row and second column might indicate the probability of moving from state 1 to state 2.
• Understanding the transition matrix allows us to predict how the system evolves over time, providing insights into steady-state probabilities.

A probability vector is a simple yet powerful concept in the realm of probability and linear algebra. It is essentially a row or a column vector where the entries are individual probabilities, each ranging from 0 to 1, and collectively summing up to 1.
This vector represents the distribution of probabilities across different possible states at a specific time or iteration. Within the scope of transition matrices, probability vectors are used to describe the current state distribution of the system. When multiplied by the transition matrix, they provide the next state probabilities.
For example, if we have a probability vector \( \begin{bmatrix} x \ y \ z \end{bmatrix} \), it means:

• "x" is the probability of being in state 1 at a given time.
• "y" is the probability of being in state 2.
• "z" is the probability of being in state 3.

The aim is often to find the steady-state probability vector, where after many transitions, the system stabilizes, and the vector remains unchanged despite further applications of the transition matrix.

In mathematical modeling, a system of linear equations arises naturally when trying to find a steady-state in Markov chains. It allows the conversion of matrix operations into simpler algebraic terms, providing a straightforward computational approach.
When searching for a steady-state vector \( \mathbf{v} \), we solve \( A\mathbf{v} = \mathbf{v} \). This can be recast into a system of equations, \( (A - I)\mathbf{v} = \mathbf{0} \), where \( I \) is the identity matrix and \( \mathbf{0} \) is the zero vector.
Breaking it down step-by-step:

• Replace the probability vector with its elements. Here \( \mathbf{v} = \begin{bmatrix} x \ y \ z \end{bmatrix} \), and each element corresponds to a state probability.
• This yields individual linear equations such as \( 0.7x + 0.1y + 0.1z = x \), which rearranges to \( -0.3x + 0.1y + 0.1z = 0 \).
• By simplifying further, we solve for one or more of these probabilities using techniques like substitution until we meet the condition that all probabilities sum up to 1.

This calculated steady-state vector offers a prediction of long-term trends for each state, assuming the transition probabilities remain constant over time.