We first consider the 
-by- Hermitian matrix 
, which has real
eigenvalues 
.
When computing eigenvalues, the following choices are possible:
for some specified set of subscripts 
,
including the special cases of the largest 
eigenvalues 
through
, and the smallest eigenvalues 
through 
.
Again, the desired accuracy may be specified.
. Again, the desired accuracy may be specified.
.
The spectral information one can request from a
singular value decomposition is similar, except that there are left and right
singular vectors and singular subspaces, corresponding to eigenvectors and
invariant subspaces. Common requests might include the numerical rank,
the null space, and the range space.
In order to compute these the user must supply a tolerance 
,
which will determine which singular values are considered to be
nonzero or zero. Methods can differ greatly depending on whether there
is a gap between the singular values larger than 
and those smaller,
and whether the user wants an approximate or precise answer.
Self-adjoint matrix pencils
, where and 
are
-by- and Hermitian, and can be simultaneously diagonalized by a
congruence, are quite similar.
The case 
corresponds to the standard Hermitian eigenproblem
just discussed.
has 
real eigenvalues which we denote
. If
is nonsingular, 
.
The user may request similar subsets of eigenvalues as described above,
as well as right and/or left eigenvectors.
If the pencil is non-singular, the user may request
right or left deflating subspaces, which are generalizations of
invariant subspaces [37].
If is a singular pencil, the
common null-space of and can also be computed,
as well as the deflating subspaces of the regular part.
Just as the SVD of A is closely related to the eigendecomposition of
,
the QSVD corresponds to the matrix pencil 
.
The case corresponds to the usual SVD.
Table 2 spells out the possible decompositions which
display all the information the user could want. We use terminology
which will apply to the non-Hermitian problem too. ``Schur'' form
refers to the simplest decomposition possible with complex unitary
transformations. (Real orthogonal matrices are also possible,
if the original data is real; we leave out the real case for ease of
exposition.) ``Jordan-Schur'' form refers to the most detailed
decomposition possible with unitary transformations; this form
reduces the matrix or matrices to ``staircase'' form [41],
wherein all the sizes of Jordan blocks are immediately available,
and some more detailed invariant
(or deflating [37] or reducing [42])
subspaces are computed. ``Jordan'' form, which typically requires
non-unitary transformations and so can be very ill-conditioned to
compute, is the most detailed decomposition. Jordan form is usually
called Weierstrass form for regular pencils, and Kronecker form
for singular pencils. 
and 
matrices in the table are unitary,
matrices are general nonsingular, 
is real and diagonal,
is real, diagonal, and nonnegative, 
matrices are
upper triangular, and 
matrices are in
upper triangular staircase form.
The decompositions may also be written with and isolated on
the left-hand-side, for example 
instead of
. The reason for writing these decompositions
as they are shown, is that they can be partial decompositions,
if only 
of the eigenvalues or singular values are of interest
to the user. In this case the central matrices (, 
,
, 
, 
, 
, 
, and 
) will be 
-by-,
and the outer matrices (, and ) will be -by-.
Table 2: The Possible ``Eigendecompositions'' of Self-adjoint Eigenproblems
In addition to these decompositions, the user may request condition numbers for any of the computed quantities (eigenvalues, means of eigenvalue clusters, eigenvectors, invariant/deflating/reducing subspaces) [13][38][27]. Given computed values for eigenvalues, eigenvectors, and/or subspaces, the user may also request an a posteriori error bound based on a computed residual.