What is a Markov chain and how would you use one?
A Markov chain is a mathematical model used to describe a sequence of events where the probability of each event depends only on the state of the previous event. In other words, it's a process where the future state of a system depends only on the current state of the system, and not on any of the previous states.

The key concept in a Markov chain is the Markov property, which states that the probability of transitioning to a new state depends only on the current state, and not on any previous states. This means that the Markov chain can be fully described by a set of probabilities for transitioning between each state.

Markov chains can be used in a variety of applications, such as in modeling stock prices, weather patterns, or even language generation. One common use of Markov chains is in natural language processing, where they can be used to generate text that follows a particular style or pattern. For example, a Markov chain model can be trained on a set of Shakespearean sonnets, and then used to generate new sonnets that follow the same patterns of rhyme and meter.

Markov chains can also be used in machine learning, particularly in reinforcement learning algorithms. In these applications, the Markov chain represents the states of an environment, and the agent learns to take actions that maximize a reward function based on the transitions between states.
Could you please provide an example of a Markov chain in matrix form and provide me with rules of how to detect if a given transition matrix can be classified as a markov chain.
```
P = [[0.8, 0.2],
     [0.3, 0.7]]
```

In this example, the Markov chain has two states, which we can label as 1 and 2. The matrix `P` represents the transition probabilities between these states. The entry `P[i][j]` represents the probability of transitioning from state `i` to state `j`.

For example, `P[1][2]` represents the probability of transitioning from state 1 to state 2, which is 0.2 in this case. Similarly, `P[2][1]` represents the probability of transitioning from state 2 to state 1, which is 0.3.

To determine if a given transition matrix can be classified as a Markov chain, we need to check if it satisfies the following two properties:

1. Non-negative entries: All entries in the matrix must be non-negative. That is, `P[i][j] >= 0` for all `i` and `j`.

2. Row sums equal to 1: The sum of the entries in each row must be equal to 1. That is, `sum(P[i][j] for j in range(n)) = 1` for all `i`, where `n` is the number of states in the Markov chain.

If a matrix satisfies these two properties, then it represents a valid Markov chain.