What is a principal component in PCA?

• A. A direction of maximum variance.
• B. The mean of the data.
• C. A random projection.
• D. The standard deviation.

Answer: (A)

In PCA, a principal component is a direction in the data space along which when you project the data, the variance becomes as large as possible. The first principal component is the axis that captures the most spread, and it can be found as the eigenvector of the data’s covariance matrix corresponding to the largest eigenvalue. This direction is a linear combination of the original variables, not just a single statistic like a mean or a standard deviation, and it’s more than a random projection because it’s chosen specifically to maximize variance. Using these directions lets you represent the data with fewer dimensions while preserving as much of the original variability as possible, with the projections onto these directions becoming the principal component scores.