Mathematics for ML

Eigenvector, explained

An eigenvector is a nonzero vector whose direction stays the same under a linear transformation.
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Part of the Semantic Notion technical glossary.

Short Definition

An eigenvector is a nonzero vector whose direction stays the same under a linear transformation.

Intuition

It is a special direction that a matrix stretches or shrinks without rotating away.

Technical Definition

For a matrix A, a nonzero vector v is an eigenvector if Av equals lambda v for some scalar eigenvalue lambda.

Example

Principal components in PCA are eigenvectors of a covariance matrix.

Common Misunderstandings

Eigenvectors are directions, not just arbitrary vectors.
Their meaning depends on the matrix they come from.

Same Category

More from Mathematics for ML

Cosine SimilarityCosine similarity measures how similar two vectors are by comparing their direction rather than their magnitude.Cross-EntropyCross-entropy measures how well one probability distribution matches another, often a model prediction against a target label.Dot ProductThe dot product combines two vectors into a single number that reflects their alignment and magnitude.EigenvalueAn eigenvalue is the scaling factor applied to an eigenvector by a linear transformation.EntropyEntropy measures uncertainty or information content in a probability distribution.Euclidean DistanceEuclidean distance is the straight-line distance between two points in a vector space.

Related Reading

Essays that use this concept

Mathematics for MLEigenvalues and Eigenvectors for Machine LearningMathematics for MLPCA vs t-SNE vs UMAP

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