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##

Conditioning

The singular values and singular vectors of have largely the
same conditioning properties as the eigenvalues and eigenvectors of
a Hermitian matrix, as described in §2.2.5.

The singular values of are always well-conditioned,
in the sense that changing in norm by at most can change
any eigenvalue by at most .

This is adequate for most purposes, unless the user is
interested in the leading digits of a small singular value, one
less than or equal to in magnitude. For example, computing
to within plus or minus
means
that no leading digits of the computed may be correct.
See [114,118] on the discussion of the sensitivity of small
singular values and of when their leading digits may be computed accurately.

Singular vectors and singular subspaces, on the other hand, can be
ill-conditioned.
The example in §2.2.5 illustrates this point.
(The eigenvalues and eigenvectors of that matrix are identical to its
singular values and singular vectors.)
Thus, the condition number of a singular vector depends on the *gap*
between its singular value and the closest other singular value;
the smaller the gap, the more sensitive the singular vector.
When singular vectors corresponding to a cluster of close singular values
are too ill-conditioned, the user may want to
compute a basis of the singular
subspace they span instead of individual singular vectors.

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