difficulty 3/5
Eigenvectors and eigenvalues
The directions a matrix does not turn, and how much it stretches each of them. Almost every method that reduces, ranks or propagates through a linear map is a question about these.
Intuition
A matrix does two things to a vector at once: it turns it and it stretches it. Almost every direction gets both. An eigenvector is a direction that gets only the second — the output lies on the same line as the input, and the eigenvalue is the factor [1, §22.2].
That sounds like a curiosity until you notice what it buys. If the matrix leaves certain directions pointing where they were, then in those directions it is just a number, and a matrix problem becomes an arithmetic one. Raising a matrix to the hundredth power is hard; raising a number to the hundredth power is not [1, §22.2.3].
Formal definition
How it works
Where it's used
Why & when
Worked example
Pitfalls
Check yourself
3 questions
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Sources
- Dive into Deep Learningbook · retrieved 2026-09-12https://d2l.ai/chapter_appendix-mathematics-for-deep-learning/eigendecomposition.html
- A Tutorial on Principal Component Analysis (2014)paperhttps://ar5iv.labs.arxiv.org/html/1404.1100
- Introduction to Information Retrieval (2008)bookhttps://nlp.stanford.edu/IR-book/html/htmledition/irbook.html
- Deep Learning (2016)bookhttps://www.deeplearningbook.org/