--- title: Eigenvalues & Eigenvectors description: Eigenvalues and eigenvectors are scalar factors and corresponding vectors that remain in the same direction during a linear transformation. You will understand how to calculate them and apply them to systems of differential equations, principal component analysis, and stability analysis. category: mathematics subcategory: linear-algebra difficulty: beginner, intermediate, advanced url: /subject/eigenvalues-and-eigenvectors --- # Eigenvalues & Eigenvectors Eigenvalues and eigenvectors are scalar factors and corresponding vectors that remain in the same direction during a linear transformation. You will understand how to calculate them and apply them to systems of differential equations, principal component analysis, and stability analysis. ## Available Resources 2 Videos • 4 Courses • 2 Websites ## Websites ### 1. Khan Academy Eigenvalues Sal Khan's video sequence introducing eigenvalues and eigenvectors, from the defining equation through the characteristic polynomial to worked 2x2 and 3x3 examples and eigenbases. Viewers finish able to compute eigenvalues, find eigenspaces, and see why eigenbases simplify transformations. **Difficulty:** Beginner | **Price:** Free **Link:** https://www.khanacademy.org/math/linear-algebra/alternate-bases/eigen-everything **Tags:** eigenvalues, eigenvectors, linear-algebra, characteristic-polynomial, eigenbasis ### 2. Wolfram MathWorld **Author:** Eric W. Weisstein MathWorld is an online mathematics encyclopedia from Wolfram Research offering detailed, browsable articles on topics across the math spectrum, including algebra, geometry, calculus, and number theory. Each entry includes definitions, theorems, formulas, diagrams, worked examples, and links to further reading. **Difficulty:** Intermediate | **Language:** English | **Price:** Free **Link:** https://mathworld.wolfram.com **Tags:** mathematics-reference, encyclopedia, abstract-algebra, number-theory, geometry ## Videos ### 1. Eigenvectors and eigenvalues Chapter 14 of 3Blue1Brown's Essence of Linear Algebra series. Grant Sanderson animates eigenvectors as the directions a transformation only stretches, then derives the characteristic equation and eigenbases, giving viewers geometric intuition behind the usual determinant-based computation and diagonalization. **Difficulty:** Beginner | **Price:** Free **Link:** https://www.youtube.com/watch?v=PFDu9oVAE-g **Tags:** eigenvalues, eigenvectors, linear-algebra, linear-transformations, diagonalization ### 2. Gilbert Strang: Eigenvalues and Eigenvectors (Differential Equations and Linear Algebra video lectures) **Author:** Gilbert Strang Eight short MIT lectures by Gilbert Strang that apply eigenvalues to differential equations. Topics are diagonalization, matrix powers and Markov matrices, solving linear systems du/dt = Au, the matrix exponential, similar and symmetric matrices, and second-order systems. **Difficulty:** Intermediate | **Language:** English | **Price:** Free **Link:** https://ocw.mit.edu/courses/res-18-009-learn-differential-equations-up-close-with-gilbert-strang-and-cleve-moler-fall-2015/pages/differential-equations-and-linear-algebra/eigenvalues-and-eigenvectors/ **Tags:** eigenvalues, diagonalization, matrix-exponential, linear-systems, markov-matrices ## Courses ### 1. Essence of Linear Algebra Grasp determinants with 3Blue1Brown's "Essence of Linear Algebra" course. Visualize and understand this key linear algebra concept. **Difficulty:** Beginner | **Price:** Free **Link:** https://www.youtube.com/playlist?list=PLZHQObOWTQDPD3MizzM2xVFitgF8hE_ab **Tags:** determinants, linear-algebra, linear-transformations, geometric-intuition ### 2. Linear Algebra (MIT 18.06) **Author:** Gilbert Strang Matrix theory and linear algebra: systems of equations, elimination, vector spaces and subspaces, orthogonality and least squares, determinants, eigenvalues and positive definite matrices. Includes 35 lecture videos plus problem sets and exams with solutions, giving the fluency needed for applied mathematics, engineering and data science. **Difficulty:** Intermediate | **Price:** Free **Link:** https://ocw.mit.edu/courses/18-06-linear-algebra-spring-2010/ **Tags:** linear-algebra, vector-spaces, eigenvalues, least-squares, matrix-factorization ### 3. Linear Algebra - Foundations to Frontiers Learn the mathematics behind linear algebra and link it to matrix software development. **Difficulty:** Intermediate | **Language:** English | **Price:** Free **Link:** https://www.edx.org/learn/linear-algebra/the-university-of-texas-at-austin-linear-algebra-foundations-to-frontiers **Tags:** courses, mathematics-statistics, linear-algebra ### 4. Mathematics for Machine Learning: Linear Algebra In this course on Linear Algebra we look at what linear algebra is and how it relates to vectors and matrices. Then we look through what vectors and matrices are and how to work with them, including the knotty problem of eigenvalues and eigenvectors, and how to use these to solve problems. Finally we look at how to use these to do fun things with datasets - like how to rotate images of faces and how to extract eigenvectors to look at how the Pagerank algorithm works. Since we're aiming at data-driven applications, we'll be implementing some of these ideas in code, not just on pencil and paper. Towards the end of the course, you'll write code blocks and encounter Jupyter notebooks in Python, but don't worry, these will be quite short, focussed on the concepts, and will guide you through if you’ve not coded before. At the end of this course you will have an intuitive understanding of vectors and matrices that will help you bridge the gap into linear algebra problems, and how to apply these concepts to machine learning. **Difficulty:** Beginner | **Language:** English | **Duration:** 5 weeks of study, 2-5 hours/week | **Price:** Free **Link:** https://www.coursera.org/learn/linear-algebra-machine-learning **Tags:** courses, mathematics-statistics, linear-algebra --- *This content is part of Dantes.io - Your Treasure Map to Knowledge* *Curated by humans at Dantes.io. Personal study use welcome; republishing this curation requires permission (team@dantes.io).* View this page online: https://dantes.io/subject/eigenvalues-and-eigenvectors