The Singular Value Decomposition (SVD) states that every matrix can be diagonalized using the proper bases for the domain and range spaces. MIT professor Gilbert Strang provides an existence proof for the SVD, broken down into an overview and details for clarity. The proof shows that a matrix A can be decomposed into UΣV^T, where U and V are orthonormal matrices and Σ is a diagonal matrix with singular values. This insightful proof highlights the relationship between singular values and eigenvalues, shedding light on the structure of the SVD. Trefethen and Bau have a proof for complex matrices, but Strang’s proof is more enlightening.
https://gregorygundersen.com/blog/2018/12/20/svd-proof/