From the course: Machine Learning Foundations: Linear Algebra
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Introduction to eigenvalues and eigenvectors
From the course: Machine Learning Foundations: Linear Algebra
Introduction to eigenvalues and eigenvectors
- [Instructor] Eigenvalues and eigenvectors are the heart of eigendecomposition. That is often called eigenvalue decomposition or eigenvector decomposition. It is only defined for square matrices, and its goal is to extract pairs of eigenvalues and eigenvectors. Each eigenvalue has an associated eigenvector. Previously, we have learned that if you apply some type of transformation on an input vector, we'll get an output vector. We can write it down as A multiplied by v equals w, where A is a transformation matrix, v as an input vector, and w is the output vector. If you look at graphical representation of this equation, we can imagine that output vector w is a scaled representation of input vector v. Then we can write our equation as lambda multiplied by v equals w. Because our two equations A multiplied by v equals w and lambda multiplied by v equals w are equivalent, we can write them as a single equation. Lambda…
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