6. Matrices and Markov Chains with Python

Overview

In this chapter, we intend to foray into matrices and their applications using Python. We will look at different matrix manipulation techniques that will help us use them efficiently to build useful tools in real-world applications.

By the end of this chapter, you will understand matrices and be able to perform operations on them. You will implement one of the applications of matrices, known as Markov chains, using transition matrices and then use Markov chains and the Markov property to solve real-world problems.

Introduction

A matrix is a rectangular array of numbers or expressions arranged in rows and columns and considered as a single entity. As we are considering a matrix as a single object, if we perform an operation on it, it will apply to each of the elements within it:

Figure 6.1: A simple m × n matrix with m rows and n columns

A simple linear single-dimension array rarely suffices for the physical world we live in, and almost all attributes related to space and time require more than one dimension. Compactness is one of the main reasons for the use of matrices. A matrix is compact when it is closed and bounded or simply has its points within a fixed distance of one another. Primarily for these two reasons, matrices find applications in practically every field, including fundamental mathematical concepts, ranging from graph theory, linear transformations, and probability theory, to different branches of physics, such as quantum mechanics and electromagnetism...

Matrix Operations on a Single Matrix

In this chapter, we will study the different ways of manipulating matrices and how to implement them in Python. Understanding how a matrix works broadly means understanding the fundamentals of how 2D or multidimensional arrays work. Once we have a good understanding of the basics of 2D matrices, those interested can delve into advanced studies of matrices, which includes special types of matrices such as sparse matrices, vector spaces, eigenvalues, and eigenvectors, which can involve more than two dimensions.

Matrices in Python can be implemented using either lists or arrays. Nested lists in Python work perfectly fine, but Python has a powerful package that makes matrix implementation much easier called NumPy. SciPy is another package that helps in matrix manipulation, but it is usually more suitable for larger matrix computations. We will be using both of these modules throughout this chapter.

Basic Operations on a Matrix

It is assumed...

Operations on Multiple Matrices

We will now be performing operations between two or more matrices and see the functions that will help us to achieve that. We will be covering how to write an inverse of a matrix, logical operators, dot products, eigenvalues and eigenvectors, outer products, and the determinates of a matrix.

It should be noted that there are plenty of other things you can do with matrices, and the official NumPy documentation is a really good resource for referencing information according to the requirements of the user. Most of the topics that we are going to cover are part of the linear algebra package of the NumPy library. There are far wider applications in physics and mathematics that are beyond the scope of this chapter for each of the topics we are going to cover, but it should suffice to know that they all play a very important role in understanding matrices.

Note

For more information about the NumPy library, refer to https://docs.scipy.org/doc/numpy...

Solving Linear Equations Using Matrices

Linear equations are the foundational blocks of algebra, and anyone who has studied basic elementary mathematics knows how they work. Let's cover them briefly, and we can then see how easily they can be solved using matrices in Python.

Linear equations are typically in the form:

Figure 6.5: Formula for calculating linear equations

Here, a1, a2,.., an are the coefficients, and x1, x2,.. xn are variables.

These linear equations with two variables can be represented in a two-dimensional space graph where x is the horizontal dimension and y is the vertical dimension.

Let's take a quick example of a linear equation with two variables. Suppose the equation is y = 2x + 6. This representation is known as the slope-intercept form and has the format y = mx + c, where m is the slope and c is the y intercept of the equation.

Here, m=2 and c=6, and the line can be drawn on a graph by plotting different values...

Transition Matrix and Markov Chains

Now, we will be looking at one of the applications of matrices, which is a field of study all by itself. Markov chains make use of transition matrices, probability, and limits to solve real-world problems. The real world is rarely as perfect as the mathematical models we create to solve them. A car may want to travel from point A to B, but distance and speed prove insufficient parameters in reality. A cat crossing the street may completely alter all the calculations that were made to calculate the time traveled by the car. A stock market may seem to be following a predictable pattern for a few days, but overnight, an event occurs that completely crashes it. That event may be some global event, a political statement, or the release of company reports. Of course, our development in mathematical and computational models has still not reached the place where we can predict the outcome of each of these events, but we can try and determine the probability...

Summary

In this chapter, we were able to cover the topic of matrices, which is fundamental to a number of topics, both in mathematics and in using Python. Data science today is primarily based on the efficient use of matrices. We studied their application in the form of Markov chains and, although an important topic, there is no dearth of topics to explore that come under the umbrella of applications using matrices.

Next, we will delve into the world of statistics. We will use a more formal and systematic approach in first understanding the building blocks of statistics, then understanding the role of probability and variables, and finally tying these concepts together to implement statistical modeling.