Segre classification
Let be an
-dimensional vector space over an algebraically closed field
(say
, but not
; this special case will be dealt with separately) and let
be a linear operator from
to itself. The essential information in
can be described by the theory of the Jordan canonical form (cf. Normal form). The essential result is that if
are the distinct eigenvalues of
with multiplicities
, then
is the direct sum
where each
is
-invariant,
and, when restricted to
,
has the form
where
is the identity mapping on
and each
is nilpotent (see e.g. [a1]).
An equivalent and perhaps more familiar description of is through the matrix
representing
with respect to a particular basis of
. The same theory says that this basis may be chosen so that
takes its Jordan canonical form
![]() |
Here, each is an
-matrix with
in each diagonal position and an entry
or
in each superdiagonal position (and it is assumed that each unnamed entry in any matrix is zero). Each matrix
can then be written in the form
![]() |
where each is an
-matrix and
with
.
A shorthand way of indicating this structure for is by means of its Segre symbol (or Segre characteristic), which is
![]() |
Thus, in the Segre symbol there is a set of entries (enclosed in round brackets) for each distinct eigenvalue. If there is only one entry in a set of brackets, the brackets are usually omitted.
An obvious important application of the theory is when . For vector spaces over
the above does not apply, since
is not algebraically closed and the proof relies on the fundamental theorem of algebra (cf. also Algebra, fundamental theorem of). Of course, the Jordan forms still apply if the characteristic equation can be solved completely over
. If not, an alternative approach (the rational canonical form) is available [a2]. If a real matrix admits a complex conjugate pair of eigenvalues, this is often indicated in the Segre symbol by a pair of entries
.
One of the important modern applications of the Segre classification is the Petrov classification of gravitational fields [a3]. Another also appears in Einstein's theory and to briefly describe it the following definitions are required.
A space-time is a -dimensional connected Hausdorff manifold admitting a Lorentz metric
of signature
. On such space-times, symmetric second-order tensors are often encountered (e.g. the Ricci, Einstein and energy-momentum tensors; cf. also Tensor on a vector space). Following the success of the Petrov classification it has been found useful to classify such tensors at a point
in
by regarding them as linear mappings on the tangent space
to
at
. Since
is a real vector space one, of course, encounters the problem of closure mentioned above. However, if the tensor in question is
, then one has the eigenvector-eigenvalue problem
![]() | (a1) |
for a complex eigenvector and eigenvalue and
, respectively. It should be noted here that one could write (a1) as
, but because of the Lorentz signature of
this would not be in the standard form to which one could apply the usual theory. There are ways of handling this latter equation but the approach (a1) is more standardized. The solution to this problem can be found in detail in [a4], [a3], [a5], [a6]. It turns out that, of the potential Segre types available, the only ones possible, bearing in mind the second equation in (a1) and the signature of
, are
,
,
and
together with their degeneracies indicated, as mentioned above, by the use of round brackets. The use of this classification of such tensors is often very useful in general relativity theory (see e.g. [a7]).
References
[a1] | D.T. Finkbeiner, "Introduction to matrices and linear transformations" , Freeman (1960) |
[a2] | G. Birkhoff, S. MacLane, "A survey of modern algebra" , Macmillan (1961) |
[a3] | A.Z. Petrov, "Einstein spaces" , Pergamon (1969) |
[a4] | S.W. Hawking, G.F.R. Ellis, "The large scale structure of space-time" , Cambridge Univ. Press (1973) |
[a5] | J.F. Plebanski, "The algebraic structure of the tensor of matter" Acta Phys. Polon. , 26 (1964) pp. 963 |
[a6] | G.S. Hall, "Differential geometry" , Banach Centre Publ. , 12 , Banach Centre (1984) pp. 53 |
[a7] | D. Kramer, H. Stephani, M.A.H. MacCallum, E. Herlt, "Exact solutions of Einstein's field equations" , Cambridge Univ. Press (1980) |
Segre classification. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Segre_classification&oldid=18578