The stochastic matrix was first developed by Andrey Markov at the beginning of the 20th century, and has found use throughout a wide variety of scientific fields, including probability theory, statistics, mathematical finance and linear algebra, as well as computer science and population genetics.
n × n matrix is called a Markovmatrix if all entries are nonnegative and the sum of each column vector is equal to 1. = 1/2 2/3 is a Markovmatrix. Markovmatrices are also called stochastic matrices. Many authors write the transpose of the matrix and apply the matrix to the right of a row vector. In linear algebra we write Ap.
In this chapter, you will learn to: Write transition matrices for Markov Chain problems. Use the transition matrix and the initial state vector to find the state vector that gives the distribution after a specified number of transitions.
A stochastic matrix, also called a probability matrix, probability transition matrix, transition matrix, substitution matrix, or Markovmatrix, is matrix used to characterize transitions for a finite Markov chain, Elements of the matrix must be real numbers in the closed interval [0, 1].
This section is about two special properties of A that guarantee a stable steady state. These properties define a positive Markovmatrix, and A above is one particular example: Markovmatrix 1. Every entry of A is positive: aij > 0. 2. Every column of A adds to 1.
As you can see, computing the powers of a stochastic matrix by hand quickly becomes difficult. However, because we are dealing with a regular stochastic matrix, we can still predict what will happen after a long time.
DEFINITION 4.3 A real n nmatrix A = [a. ij] is called a Markovmatrix, or row{ stochastic matrix if (i) a. ij 0 for 1 i;j n; (ii) Pn j=1. a. ij= 1 for 1 i n. Remark: (ii) is equivalent to AJ. n= J. n, where J. n= [1;:::;1]t. So 1 is always an eigenvalue of a Markovmatrix.
Probablistic models like Markov chains are very common in game theory. In this section, I want to look at very simple games of chance (though the theory extends well to more complicated games).
A matrix satisfying conditions of (0.1.1.1) is called Markov or stochastic. Given an initial distribution P [X = i] = pi, the matrix P allows us to compute the the distribution at any subsequent time.
