▶Covariance eigendecomposition
A symmetric covariance matrix has orthogonal eigenvectors and nonnegative eigenvalues.
Principal Component Analysis: Definition
Center, diagonalize, and project
PCA turns the geometric intuition into a reproducible calculation: center the observations, form their covariance matrix, and use its eigenvectors as new axes.
watchone idea
→manipulateone example
→leave withone intuition
The essential idea: principal directions are covariance eigenvectors, ordered by their eigenvalues.
One Concept · One Example
What to notice
- Centering removes the mean before rotation.
- Eigenvectors give orthogonal directions.
- Eigenvalues quantify variance explained.
Covariance becomes a coordinate system
Projection
Centered observations are expressed in the principal-component basis.
Build PCA from the covariance matrix
Change the correlation. The covariance ellipse, eigenvalues, and explained-variance ratio update together.
PCA is an eigendecomposition of centered covariance followed by projection onto the leading eigenvectors.