Matrix Factorization and Latent Semantic Indexing SVD Singular value decomposition For an mxnmatrix A of rank r there exists a factorization Singular Value Decomposition= SVD as follows 4 MxM MxN Vis NxN The columns of u are orthogonal eigenvectors of aat The columns of v are orthogonal eigenvectors of aA Eigenvalues Ml. Mr of aaT are the eigenvalues of atA = Singular valuesMatrix Factorization and Latent Semantic Indexing 16 Singular Value Decomposition A=UV T MM MN V is NN For an M  N matrix A of rank r there exists a factorization (Singular Value Decomposition = SVD) as follows: The columns of U are orthogonal eigenvectors of AAT. The columns of V are orthogonal eigenvectors of ATA. i =i =diag (1 ...r ) Singular values. Eigenvalues 1 … r of AAT are the eigenvalues of ATA. SVD