Principal component directions
Theorem taught in Principal component analysis, part of Statistical learning
Principal component directions
| Type | Theorem |
| Taught in | Principal component analysis |
| Part | Statistical learning |
Let S=\Gamma\Lambda\Gamma^\top with \lambda_1\ge\cdots\ge\lambda_p\ge 0 and \Gamma=(\boldsymbol{\gamma}_1,\ldots,\boldsymbol{\gamma}_p) orthogonal. The jth principal component direction is \mathbf{a}_j=\boldsymbol{\gamma}_j. The jth principal component Y_j=\boldsymbol{\gamma}_j^\top\mathbf{X} has sample variance \lambda_j. Distinct components are uncorrelated, and the total variance is preserved:
\sum_{j=1}^p\operatorname{Var}(Y_j)=\sum_{j=1}^p\lambda_j=\operatorname{tr}(S)=\sum_{j=1}^p s_{jj}.
Used in
- Taught in Principal component analysis, where the derivation and examples are.
- All 7 questions on Principal component analysis
See also
Principal component directions and scores · Eigenvalues and eigenvectors · Singular value decomposition · Variance · Scaling · Principal components by Lagrange multipliers · Reconstruction from principal components · Choosing the number of principal components