= Solution
{c}
Let $X_c$ be the column-centered <data matrix>. Its sample <covariance matrix> is $S=X_c^TX_c/(n-1)$. The <loadings matrix> $V=(v_1,\ldots,v_p)$ has orthonormal <eigenvectors> of $S$ as columns, ordered by nonincreasing <eigenvalues>:
$$
Sv_j=\lambda_jv_j,\qquad V^TV=I.
$$
Each <principal component loading> specifies the coefficients of a principal direction in the original variables. Signs are arbitrary, and equal <eigenvalues> permit orthogonal changes of basis within their <eigenspaces>.
Back to article page
