Math 060 Fall 2017 120617C Singular Value Decomposition Part 2

Review of the compact singular value decomposition. Recall the cast of characters: V; V_1, S_1, U_1. Constructing the Singular Value Decomposition of a matrix A: first observe that U_1 has orthonormal columns that form an orthonormal basis of R(A); use Gram-Schmidt to extend those columns to a full orthonormal basis (and thus U_1 to an orthogonal matrix U). Claim: A = U S V^T. Singular vectors: definition (right singular vectors, left singular vectors). Relation between the singular vectors. Outer product expansion of A (using the singular vectors). Constructing a lower rank approximation of A using the outer product expansion. Theorem (statement only): such a low-rank approximation is the "closest" (w.r.t. the Frobenius norm) to A. End of course.