The Eckart-Young Theorem

For any mxn matrix (operator acting on Euclidean space) we prove that the best finite rank approximation of this matrix in the norm sense is given by the truncated singular value decomposition. This is the Eckart-Young Theorem. This theorem also asserts that the smallest possible operator norm distance between the operator and a finite rank operator of rank l is the (l+1)st singular value. #mikethemathematician, #mikedabkowski, #profdabkowski