Section 9.1 introduces the basic ideas and technical elements behind principal components. No particular assumption will be made on except that the mean vector and the covariance matrix exist. When reference is made to a data matrix in Section 9.2, the empirical mean and covariance matrix will be used. Section 9.3 shows how to interpret the principal components by studying their correlations with the original components of . Often analyses are performed in practice by looking at two-dimensional scatterplots. Section 9.4 develops inference techniques on principal components. This is particularly helpful in establishing the appropriate dimension reduction and thus in determining the quality of the resulting lower dimensional representations. Since principal component analysis is performed on covariance matrices, it is not scale invariant. Often, the measurement units of the components of are quite different, so it is reasonable to standardize the measurement units. The normalized version of principal components is defined in Section 9.5. In Section 9.6 it is discovered that the empirical principal components are the factors of appropriate transformations of the data matrix. The classical way of defining principal components through linear combinations with respect to the largest variance is described here in geometric terms, i.e., in terms of the optimal fit within subspaces generated by the columns and/or the rows of as was discussed in Chapter 8. Section 9.9 concludes with additional examples.