lecture20

lecture20 - Lecture 20 SVD and Its Applications Shang-Hua...

This preview shows pages 1–7. Sign up to view the full content.

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
This is the end of the preview. Sign up to access the rest of the document.

Unformatted text preview: Lecture 20 SVD and Its Applications Shang-Hua Teng Every symmetric matrix A can be written as Spectral Theorem and Spectral Decomposition [ ] T n n n T T n T n n x x x x x x x x A λ λ λ λ + + = = 1 1 1 1 1 1 where x 1 … x n are the n orthonormal eigenvectors of A, they are the principal axis of A. Singular Value Decomposition • Any m by n matrix A may be factored such that A = U Σ V T • U : m by m , orthogonal, columns • V : n by n , orthogonal, columns 2200 Σ : m by n , diagonal, r singular values The Singular Value Decomposition r = the rank of A = number of linearly independent columns/rows A U V T m x n m x m m x n n x n · · = Σ SVD Properties • U , V give us orthonormal bases for the subspaces of A : – 1st r columns of U : Column space of A – Last m - r columns of U : Left nullspace of A – 1st r columns of V : Row space of A – 1st n- r columns of V : Nullspace of A • IMPLICATION: Rank( A ) = r The Singular Value Decomposition · · A U V T = Σ A U V T m x n m x r r x r r x n = Σ m x n m x m m x n n x n Singular Value Decomposition...
View Full Document

{[ snackBarMessage ]}

Page1 / 17

lecture20 - Lecture 20 SVD and Its Applications Shang-Hua...

This preview shows document pages 1 - 7. Sign up to view the full document.

View Full Document
Ask a homework question - tutors are online