Math 060 Fall 2017 120417 Singular Value Decomposition

Review of various facts regarding A^T A. Definition of singular value decomposition. Theorem: every matrix has a singular value decomposition. Proof by construction: Step I (Constructing the compact SVD). Observations: A^T A has real, non-negative eigenvalues. A^T A is orthogonally diagonalizable (by some matrix V). The matrix S_1 of singular values. Splitting V into a matrix V_1 whose columns form an orthonormal basis of R(A^T) and another V_2 whose columns an orthonormal basis of N(A). Using S_1 and V_1 to express A as a compact singular value decomposition.