Matrix Factorization and Latent Semantic Indexing Background Eigenvalues Eigenvectors For symmetric matrices, eigenvectors for distinct eigenvalues are orthogonal All eigenvalues of a real symmetric matrix are real All eigenvalues of a positive semidefinite matrix are non-negative 8Matrix Factorization and Latent Semantic Indexing 8 Eigenvalues & Eigenvectors Sv{1,2}={1,2}v{1,2} ,and 121•v2=0 For symmetric matrices, eigenvectors for distinct eigenvalues are orthogonal All eigenvalues of a real symmetric matrix are real.  n ,w TSw0, then ifSv=v0 All eigenvalues of a positive semidefinite matrix are non-negative Background