Diffusion maps are a coordinate transformation method with the goal of learning nonlinear manifolds in a high dimensional ambient space. The main idea behind this method is to redefine the distance between our data points, using a measure which will incorporate all available data points instead of focusing simply on the two points of interest, so that the global structure can be reflected locally. With this new measure of distance, we transform the coordinates so that the standard euclidean distance equates to newly defined distance measure. In this paper we present the probabilistic interpretation of diffusion maps based on Markov chains, specifically, random walks on a graph. For a given dataset, by using a diffusion kernel, we define a transition matrix for a random walk on a graph, for which we calculate corresponding eigenvalues and eigenvectors needed to transform starting coordinates to diffusion coordinates. The eigenproblem proves to be too challenging for exact calculation, which compels us to use numerical methods for eigenvalues, with symmetrical matrices as our main focus, which in the end gives us an efficient and precise algorithm for calculating diffusion maps. With our algorithm in hand, we apply it to problems of classification and segmentation. For classification we compare our algorithm to a very popular and widely used algorithm known as PCA, and we also compare it to a very theoretically similar method to diffusion maps known as Locally Linear Embedding method. When it comes to segmentation, the problem of data size arises, with which we deal with using Nyström approximation, so that finally we obtain an algorithm which provides decent results.