In principal component analysis, eigenvectors and eigenvalues explain variance; Which statement about PCA is accurate?

• A. Eigenvectors define directions that remain invariant under the transformation; eigenvalues give the scaling; PCA uses the top eigenvectors.
• B. Eigenvectors define rotation axes; eigenvalues are means of the data; PCA uses the eigenvectors with smallest eigenvalues.
• C. Eigenvectors are the basis of the original coordinates; eigenvalues are random; PCA uses random components.
• D. Eigenvectors maximize distances; eigenvalues measure skew; PCA uses all components.

Answer: (A)

In PCA, you look at the covariance structure of centered data and diagonalize it. The eigenvectors of the covariance matrix point in directions where the data’s variance is aligned; they are the axes that rotate the data into a new coordinate system. The corresponding eigenvalues tell you how much variance lies along each of those directions. So the principal components are the projections onto those eigenvectors, and the amount of variance explained by each component is given by its eigenvalue.

Because you want to capture as much variance as possible with fewer dimensions, you typically select the eigenvectors associated with the largest eigenvalues—the top directions of greatest variance—and project the data onto them. That makes the statement in question correct: eigenvectors define invariant directions under the transformation (they map to a scalar multiple of themselves), eigenvalues quantify the scaling along those directions (the explained variance), and PCA uses the top eigenvectors to form a lower-dimensional representation.

The other descriptions don’t fit PCA. Eigenvalues are not means of the data, and PCA doesn’t rely on the smallest eigenvalues for the reduced representation. Eigenvectors aren’t simply the basis of the original coordinates—their role in PCA is to define new, variance-maximizing axes. And while PCA can consider all components, the common practice for dimensionality reduction is to use the top components, not all of them.