Presentation on theme: "Lecture 19 Singular Value Decomposition"— Presentation transcript:
1 Lecture 19 Singular Value Decomposition Shang-Hua Teng
2 Spectral Theorem and Spectral Decomposition Every symmetric matrix A can be written aswhere x1 …xn are the n orthonormal eigenvectors of A, they arethe principal axis of A.xi xiT is the projection matrix on to xi !!!!!
3 Matrix DecompositionDoes every matrix, not necessarily square matrix, have a similar decomposition?How can we use such a decomposition?
4 Singular Value Decomposition Any m by n matrix A may be factored such thatA = UVTU: m by m, orthogonal, columnsV: n by n, orthogonal, columns: m by n, diagonal, r singular values
6 The Singular Value Decomposition VTm x n m x m m x n n x n=Sr = the rank of A= number of linearly independentcolumns/rows
7 The Singular Value Decomposition VT=m x n m x m m x n n x nr = the rank of A= number of linearly independentcolumns/rows
8 SVD Properties U, V give us orthonormal bases for the subspaces of A: 1st r columns of U: Column space of ALast m - r columns of U: Left nullspace of A1st r columns of V: Row space of A1st n - r columns of V: Nullspace of AIMPLICATION: Rank(A) = r
9 The Singular Value Decomposition AUSVT=m x n m x m m x n n x nAUSVT=m x n m x r r x r r x n
10 The Singular Value Decomposition =SVTm x n m x m m x n n x nAU=SVTm x n m x r r x r r x n
11 Singular Value Decomposition whereu1 …ur are the r orthonormal vectors that are basis of C(A) andv1 …vr are the r orthonormal vectors that are basis of C(AT )