This tool will help you calculate the resulting state probabilities from a given transition matrix.

## How to Use the Transition Matrix Calculator

This calculator is designed to help you easily calculate the next state probabilities given a transition matrix. Follow these steps to use the calculator:

- Enter the number of states in the system.
- Click “Generate Matrix Input” to create input fields for your transition matrix.
- Fill in each cell of the transition matrix with the corresponding probability value.
- Click “Calculate” to see the result.

### How It Calculates Results

The transition matrix calculator uses the provided matrix of probabilities to calculate the next state probabilities. Each entry in the matrix represents the probability of transitioning from one state to another.

The result is computed using matrix multiplication of the transition matrix with the state vector. This gives the probabilities of the system being in each state after one transition step.

### Limitations

The calculator assumes that the transition matrix is valid (i.e., each row sums to 1) and that the probabilities are properly normalized. Ensure your inputs are correct to get reliable results.

## Use Cases for This Calculator

### Calculate the next state probabilities

Enter the transition matrix values for different states in the calculator to determine the probabilities of transitioning to each state. This allows you to understand the likelihood of moving from one state to another based on the given transition matrix.

### Verify if the matrix is stochastic

By inputting the values for the transition matrix, you can check if the matrix is stochastic, ensuring that each row’s elements sum up to 1. This feature helps you validate the accuracy of the matrix for further calculations.

### Evaluate the steady-state vector

Use the calculator to find the steady-state vector by solving the equation (πQ = 0, π1 = 1), where π represents the stationary distribution and Q is the sub-matrix without the identity matrix. This allows you to determine the long-term probabilities for each state.

### Analyze the transition probabilities

Input the transition matrix values to analyze the probabilities of moving between states over multiple time steps. This feature helps in forecasting future state probabilities in a Markov chain process.

### Determine the absorbing states

Utilize the calculator to identify absorbing states by evaluating the matrix and spotting rows where a state transitions only to itself with a probability of 1. This assists in understanding states from which there is no escape within the system.

### Calculate the expected number of steps to absorption

Enter the transition matrix values to compute the expected number of steps to absorption in the absorbing Markov chain. This calculation provides insights into the average time required to reach absorbing states from any initial state.

### Visualize the state transition diagram

After inputting the transition matrix values, generate a visual representation of the state transition diagram to understand the flow of transitions between different states visually. This feature enhances the comprehension of the system dynamics.

### Simulate the Markov chain process

Simulate the Markov chain process by inputting initial state probabilities and transition matrix values to observe how the system evolves over multiple time steps. This simulation helps in predicting future states based on the current state.

### Calculate the hitting probabilities

By entering the transition matrix values, calculate the hitting probabilities to determine the likelihood of reaching a specific state starting from any initial state. This analysis aids in understanding the accessibility of different states within the system.

### Identify the communicating classes

Utilize the calculator to identify the communicating classes within the Markov chain by analyzing the matrix and grouping states that can reach each other eventually. This classification helps in understanding the interconnections between states in the system.