LaTeX
\Sigma v_k=\lambda_k v_kVariables
- Σ: covariance matrix
- vk: principal direction
- λk: explained variance
How to use this formula
Finds principal directions as covariance-matrix eigenvectors.
Important notes
- Center variables before forming the covariance matrix.
Quick example
Order components by decreasing eigenvalue.
Applicability, worked calculation, and verification
Assumptions and domain checks
- Center variables before forming the covariance matrix.
- For the PCA Eigenvalue Problem, identify whether each quantity is a sample statistic, population parameter, estimator, or model value, and check the method assumptions.
Worked example
Order components by decreasing eigenvalue.
Common mistakes
- Do not substitute sample and population quantities interchangeably in PCA Eigenvalue Problem; map every symbol to its definition first.
- For the PCA Eigenvalue Problem, do not interpret a descriptive statistic as a causal or population conclusion without the sampling and model assumptions.
Continue the workflow
Use PCA Eigenvalue Problem in your own work
- Check the domainMatch the variables and assumptions to the problem before substituting values.
- Copy the exact notationPreserve grouping, signs, and exponents in
\Sigma v_k=\lambda_k v_k. - Edit or convertOpen the expression in the LaTeX editor, then export it for your document or web page.
Review and verification
Last reviewed: 2026-07-23
Automated quality check: Kept noindex until the missing evidence is supplied.
Formula references
- Introductory Statistics 2eOpenStax, Rice University — Reviewed probability and statistics definitions, notation, and formulas.
Frequently asked questions
What is the PCA Eigenvalue Problem used for?
Finds principal directions as covariance-matrix eigenvectors.
Can I copy this formula as LaTeX?
Yes. Copy \Sigma v_k=\lambda_k v_k or open it in the LaTeX editor.
What should I check before using it?
Confirm that each variable, unit, domain restriction, and assumption matches the problem.