We study the fairness of dimensionality reduction methods for recommendations. We focus on the fundamental method of principal component analysis (PCA), which identifies latent components and produces a low-rank approximation via the leading components while discarding the trailing components. Prior works have defined notions of "fair PCA"; however, these definitions do not answer the following question: why is PCA unfair? We identify two underlying popularity mechanisms that induce item unfairness in PCA. The first negatively impacts less popular items because less popular items rely on trailing latent components to recover their values. The second negatively impacts highly popular items, since the leading PCA components specialize in individual popular items instead of capturing similarities between items. To address these issues, we develop a polynomial-time algorithm, Item-Weighted PCA, that flexibly up-weights less popular items when optimizing for leading principal components. We theoretically show that PCA, in all cases, and Normalized PCA, in cases of block-diagonal matrices, are instances of Item-Weighted PCA. We empirically show that there exist datasets for which Item-Weighted PCA yields the optimal solution while the baselines do not. In contrast to past dimensionality reduction re-weighting techniques, Item-Weighted PCA solves a convex optimization problem and enforces a hard rank constraint. Our evaluations on real-world datasets show that Item-Weighted PCA not only mitigates both unfairness mechanisms, but also produces recommendations that outperform those of PCA baselines.