A Comparative Analysis of Principal Component Analysis (PCA) and Singular Value Decomposition (SVD) as Dimensionality Reduction Techniques

High-dimensional image data often require dimensionality reduction before further analysis. This paper provides a purely analytical comparison of two linear techniques-Principal Component Analysis (PCA) and Singular Value Decomposition (SVD). After the derivation of each algorithm from first principles, we assess their interpretability, numerical stability, and suitability for differing matrix shapes. We synthesize rule-of-thumb guidelines for choosing one out of the two algorithms without empirical benchmarking, building on classical and recent numerical literature. Limitations and directions for future experimental work are outlined at the end.