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## Related Singular Value Problems

1. If , and the SVD of , then the SVD of .

2. Suppose , where is by with . Then is an by Hermitian matrix. Let be the SVD of . Then the eigendecomposition of . Note that . In other words, the eigenvectors of are the right singular vectors of , and the eigenvalues of are the squares of the singular values of . If , where is by with , then is by with eigendecomposition . In other words, the eigenvectors of are the right singular vectors of , and the eigenvalues of are the squares of the singular values of , in addition to 0 with additional multiplicity .

3. Suppose , where is by with . Then is an by Hermitian matrix. Let be the SVD of . Write and . Then the eigendecomposition of , where and

In other words, is an eigenvalue with unit eigenvector for to , and 0 is an eigenvalue with eigenvector for .

4. The QSVD or generalized SVD (GSVD) of and is defined as follows. Suppose is by and is by and nonsingular. Let . Let the SVD of . We may also write two equivalent decompositions of and as and , where is by and nonsingular, is by and contains in its leading rows and columns and zeros elsewhere, and is by and contains .
and  can be chosen so that . Thus . The diagonal entries of are called the generalized singular values of and .

This decomposition generalizes to the cases where is singular or by , to a decomposition equivalent to the SVD of (the product SVD), and indeed to decompositions of arbitrary products of the form  [108].

5. The CS (cosine/sine) decomposition of and is defined as follows. Suppose the columns of are orthonormal. The QSVD of and is then equivalent to the decompositions and , where is unitary. The diagonal entries of () are the cosines (sines) of the principal angles between the subspace spanned by the columns of and the subspace spanned by the first columns of [424,25]. This is used to compute the distance'' (or angles) between subspaces.

6. Suppose is -by-, and is by with full column rank; then the GHEP can be solved using the QSVD as follows. Write the QSVD as and . The eigendecomposition of may be written and . This generalizes to the case of without full column rank.

7. Finding that minimizes is the linear least squares problem. Suppose is by with ; we say that the least squares problem is overdetermined. Let be the compact SVD of . If has full rank, the unique solution of the least squares problem is , although solutions based on the QR decomposition or normal equations are cheaper and more commonly used [50,12]. If 's rank is less than , then the SVD is often used to solve the least squares problem. In this case the solution is not unique but its unique minimum norm solution is

If we say that the least squares problem is underdetermined. In this case the solution is not unique but its unique minimum norm solution is , where is the compact SVD of .

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