A Bottom-Up Inductive Proof of the Singular Value Decomposition
C.-T. Pan, Kermit Sigmon · SIAM Journal on Matrix Analysis and Applications · 1994
The singular value decomposition (SVD) has a long history. The first proofs of the SVD for real square matrices came out of the study of bilinear forms, first by Beltrami in 1873 and, independently, by Jordan in 1874. Beltrami recognized and used the relationship of the SVD to the eigenvalue decomposition of the matrices $A^T A$ and $AA^T $, while Jordan used an inductive argument that constructs the SVD from the largest singular value and its associated singular vectors. Many proofs of the SVD in modern references are still based on one of these methods. The purpose of this note is to give a new simple “bottom-up” inductive proof of the SVD, starting from the smallest singular value, which is essentially different from either of these methods.