SVD and PCA: How Linear Algebra Compresses Thousands of Dimensions

SVD and PCA reduce a million-column matrix to ten while preserving most information, powering recommendation systems, image compression, and machine learning. A new arXiv book by Bandeira, Singer, and Strohmer dedicates a chapter to these techniques, which this article explains from scratch with Python code. The Eckart-Young theorem proves that keeping the top k singular values yields the best rank-k approximation mathematically.