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 .
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
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
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.