The symmetry of and
is purely an algebraic property and is
not sufficient to ensure any of the special
mathematical properties enjoyed by a definite matrix pencil, such
as those discussed in §2.3.
In fact, it can be shown that any real square matrix
may be written
as
or
, where
and
are suitable
symmetric matrices; for example, see [353].
The eigenvalues of a definite matrix pencil are all real,
but an indefinite pencil
may have complex eigenvalues. For example, when
In §2.3, we know that
when is positive definite we can find a matrix of eigenvectors
and a diagonal matrix of eigenvalues
such that
with
. The
inner product
forms a true
inner product and
is a norm.
When is indefinite and nonsingular
and if
is not defective (i.e., no eigenvectors are missing),
we can find a full set of eigenvectors,
. The equation
still holds, and the eigenvectors can be chosen so that
, where
is a diagonal matrix with
and
on the diagonal
(note that it is transpose, not conjugate transpose,
even though some vectors can be complex).
The
inner product
is an indefinite inner product or
pseudo-inner product, and
can be used for normalizing purposes. Unlike the positive definite case, there
is a set of vectors having pseudolength zero (as measured by
).
In fact, it is possible for an eigenvector
to satisfy
. This implies that the Rayleigh quotient
The analogy between the definite and the indefinite cases
can be taken further. When is positive definite,
is an approximate
eigenvector, and
is an approximate eigenvalue, we have the standard
residual bound:
If both and
are singular,
or close to singular, worse problems may occur.
Assume that there is a
nonzero vector
such that
; then any complex number
is an eigenvalue.
A more general case is illustrated in the example
below:
In practice, if
we have a singular pencil, is singular for any
,
and a good LU factorization routine should give a warning (when used
in (a) and (b) in the following §8.6.2).
Another sign of a singular pencil is that the eigenvalue routine produces
some random eigenvalues at each run on the same problem.