Question

Linear algebra thank you.

1. Explain in your own words how eigenvalues and eigenvectors can be useful when finding the 50th cycle (or itteration) of a Markov chain.

Answer 1

The Markov chain can be solved numerically by using the Power Methid which is an iterative method.

The power method can be used to find the stationery states of these Markov Chain. The eigen values and corresponding eigen vectors can be found by applying this method. Each and every eigen value can be found by Power method. The condition required is that the matrix of any order should be diagonalisable matrix. We can find smallest and largest value of eigen value and eigen vector.

The power method is applied on nP=P type of matrix equation where P is a transition matrix generally but it can be applied on other matrix also. In the case transition matrices the dominant eigen value is 1 but for matrices other than transition matrices dominant eigen value can be other than 1. The general interpertation of eigen values using Power method is given in image below.

After seeing the images you can understand how Power method helps us to find the dominant or largest eigen value and eigen vector by a large number of iteration with required precision.