Introduction To Eigenvalues And Eigenvectors With Numpy

Leo Migdal

-Nov 17, 2025, 3:38 PM

introduction to eigenvalues and eigenvectors with numpy

Welcome to the first lesson of our course on Linear Algebra with NumPy. In this lesson, we will explore the concepts of eigenvalues and eigenvectors. These concepts play a critical role in various fields such as engineering and data science. We'll focus on applying these concepts practically, without delving into the underlying mathematical theory. By the end of this lesson, you'll be ready to use NumPy to compute eigenvalues and eigenvectors of your matrices. Let's start delving into the process, by breaking it down into clear, manageable steps.

In linear algebra, eigenvalues and eigenvectors are central to understanding how linear transformations affect vector spaces. Eigenvectors remain in their span (retain their direction) when a transformation is applied, merely being scaled by their corresponding eigenvalue—this scalar factor determines the degree of stretching or compression. This characteristic simplifies the analysis of transformations, allowing complex matrices to be broken down into simpler components. Such decomposition is vital for grasping the core structure of a transformation, helping us to interpret its fundamental behavior without unnecessary complexity. First, we need to define a square matrix. A square matrix has the same number of rows and columns.

Here's how you do it in NumPy: This code creates a 2x2 matrix with predefined values. Compute the eigenvalues and right eigenvectors of a square array. Matrices for which the eigenvalues and right eigenvectors will be computed The eigenvalues, each repeated according to its multiplicity. The eigenvalues are not necessarily ordered.

The resulting array will be of complex type, unless the imaginary part is zero in which case it will be cast to a real type. When a is real the resulting eigenvalues will be real (0 imaginary part) or occur in conjugate pairs The normalized (unit “length”) eigenvectors, such that the column eigenvectors[:,i] is the eigenvector corresponding to the eigenvalue eigenvalues[i]. If the eigenvalue computation does not converge. In the realm of linear algebra, eigenvectors and eigenvalues play a pivotal role in understanding the properties of linear transformations. These concepts find extensive applications in various fields such as computer science, physics, engineering, and data analysis.

NumPy, a powerful Python library for numerical computing, provides a convenient way to compute eigenvectors and eigenvalues. In this blog post, we will explore the fundamental concepts of eigenvectors in NumPy, learn how to use the relevant functions, look at common practices, and discover some best practices for efficient usage. Let (A) be an (n\times n) square matrix. A non - zero vector (\mathbf{v}) is called an eigenvector of (A) if there exists a scalar (\lambda) such that: The scalar (\lambda) is called the eigenvalue corresponding to the eigenvector (\mathbf{v}). Geometrically, an eigenvector of a matrix (A) is a vector that only changes by a scalar factor when the matrix (A) is applied to it.

NumPy provides the numpy.linalg.eig function to compute the eigenvalues and right eigenvectors of a square array. Here is a simple code example: In this code, we first define a (2\times2) square matrix A. Then we use the np.linalg.eig function to compute the eigenvalues and eigenvectors of A. The function returns two arrays: the first array contains the eigenvalues, and the second array contains the corresponding eigenvectors. Each column of the eigenvectors array corresponds to an eigenvector.

In linear algebra, eigenvalues and eigenvectors play a crucial role in decomposing matrices, solving systems of equations, and even powering algorithms in machine learning and computer graphics. If you're new to this concept, think of eigenvectors as special directions that remain unchanged (except for scaling) when a transformation (matrix) is applied. The eigenvalue tells us how much the vector gets scaled. And in this tutorial, you’ll learn to compute and interpret them using NumPy. We're working with a 2x2 matrix here. Eigen decomposition is only defined for square matrices, so always verify the matrix shape using:

Explanation: np.linalg.eig() returns two results: Understanding eigenvalues and eigenvectors is key to mastering linear algebra concepts for scientific computing and data analysis. This comprehensive guide will explain eigenvalues and eigenvectors in depth, and demonstrate how to calculate them in Python with NumPy. Read on to gain true expertise in this fundamental area of matrix algebra! Eigenvalues and eigenvectors have important meanings related to matrix transformations. For a square matrix A, the eigenvalues are scalar solutions to:

Where I is the identity matrix. The solutions λ are the eigenvalues of A. Geometrically, these represent scale factors in matrix transformations. So eigenvectors are non-zero vectors that only change by a scalar λ when transformed by the matrix. In this tutorial, we will explore NumPy's numpy.linalg.eig() function to deduce the eigenvalues and normalized eigenvectors of a square matrix. In Linear Algebra, a scalar $\lambda$ is called an eigenvalue of matrix $A$ if there exists a column vector $v$ such that

and $v$ is non-zero. Any vector satisfying the above relation is known as eigenvector of the matrix $A$ corresponding to the eigen value $\lambda$. We take an example matrix from a Schaum's Outline Series book Linear Algebra (4th Ed.) by Seymour Lipschutz and Marc Lipson1. $$ A = \begin{bmatrix} 3 & 1 \\ 2 & 2 \end{bmatrix}, $$ Let $A$ be an $n\times n$ matrix (i.e. a square matrix).

A non-zero vector $\vec{v}$ is an eigenvector of $A$ with eigenvalue $\lambda$ if Rewriting this equation, we see that $\vec{v}$ is a solution of the homogeneous system of equations where $I$ is the identity matrix of size $n$. Non-trivial solutions exists only when the matrix $A-\lambda I$ is noninvertible (singular). That is, when $\operatorname{det}(A-\lambda I) =0$. Therefore, the eigenvalues are the roots of the characteristic polynomial

Here are three examples that we will consider. In each case, we have pre-computed the eigenvalues and eigenvectors (check them yourself). Notice, for matrix $D$ there is one eigenvalue that has two associated eigenvectors. Linear algebra is the backbone of countless modern technologies, from machine learning algorithms to complex engineering simulations. Among its most fundamental concepts are eigenvalues and eigenvectors. These special numbers and vectors reveal intrinsic properties of linear transformations, offering profound insights into the behavior of systems.

While the underlying mathematics can seem daunting, Python”s powerful NumPy library makes calculating and understanding eigenvalues and eigenvectors surprisingly straightforward. In this post, we”ll demystify these concepts and show you how to leverage NumPy for efficient computation. Imagine a linear transformation — like stretching, rotating, or shearing — applied to a vector. For most vectors, both their direction and magnitude will change. However, there are special vectors, called eigenvectors, that only get scaled by the transformation, without changing their direction (they might point in the opposite direction, but that”s still along the same line). This relationship is captured by the equation: Av = λv

Essentially, when you multiply the matrix A by its eigenvector v, the result is simply a scaled version of the same eigenvector v, where the scaling factor is the eigenvalue λ. © 2025 ApX Machine LearningEngineered with @keyframes heartBeat { 0%, 100% { transform: scale(1); } 25% { transform: scale(1.3); } 50% { transform: scale(1.1); } 75% { transform: scale(1.2); } }

Introduction To Eigenvalues And Eigenvectors With Numpy

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