Eigenvalues & Eigenvectors Introduction

Linear Algebra for Machine Learning · Axiom Academy

The directions that don't change Imagine a transformation—a matrix multiplication that stretches, rotates, and shears the coordinate plane. Most vectors spin around and point in different directions after this transformation. But there are special vectors that do something different: they only get scaled. These special vectors are eigenvectors . An eigenvector is a non-zero vector that, when multiplied by a matrix, points in the same direction (or exactly opposite) but gets stretched or compressed by a constant factor called the eigenvalue . Below, watch what happens when we apply a matrix transformation. The green eigenvector stays in place while the red vector rotates: Most vectors change direction when multiplied by a matrix. Their orientation and magnitude both shift in complex ways. These special vectors keep their direction (or reverse). They only scale by a constant factor called the eigenvalue. If v is an eigenvector of matrix A with eigenvalue λ, then: This elegant equation says: "When we multiply the matrix A by the eigenvector v, we get the same vector back, just scaled by λ." Machine Learning: PCA uses eigenvectors to find the most important directions in data. Physics: Eigenvectors describe natural vibration modes of structures. Search Engines: Google's PageRank algorithm is essentially finding the dominant eigenvector of the web graph. Stability Analysis: Eigenvalues tell us if systems grow or decay over time.

This is the written version of the interactive lesson above. See the full Linear Algebra for Machine Learning course.