Next: Singular Subspaces Up: Singular Value Decomposition  J. Previous: Singular Value Decomposition  J.   Contents   Index

## Singular Values and Singular Vectors

The square roots of the eigenvalues of are the singular values of . Since is Hermitian and positive semidefinite, the singular values are real and nonnegative. This lets us write them in sorted order .

The eigenvectors of are called (right) singular vectors. We denote them by , where is the eigenvector for eigenvalue . The by matrix is also Hermitian positive semidefinite. Its largest eigenvalues are identical to those of , and the rest are zero. The eigenvectors of are called (left) singular vectors. We denote them by , where through are eigenvectors for eigenvalues through , and through are eigenvectors for the zero eigenvalue. The singular vectors can be chosen to satisfy the identities and for , and for .

We may assume without loss of generality that each and . The singular vectors are real if is real. Though the singular vectors may not be unique (e.g., any vector is a singular vector of the identity matrix), they may all be chosen to be orthogonal to one another: if . When a singular value is distinct from all the other singular values, its singular vectors are unique (up to multiplication by scalars).

Next: Singular Subspaces Up: Singular Value Decomposition  J. Previous: Singular Value Decomposition  J.   Contents   Index
Susan Blackford 2000-11-20