Segre classification

Let $V$ be an $n$-dimensional vector space over an algebraically closed field $F$ (say $F=C$, but not $\textbf{R}$; this special case will be dealt with separately) and let $f$ be a linear operator from $V$ to itself. The essential information in $f$ can be described by the theory of the Jordan canonical form (cf. Normal form). The essential result is that if $\lambda_1,...,\lambda_r$ are the distinct eigenvalues of $f$ with multiplicities $n_1,...,n_r$, then $V$ is the direct sum $V=V_1\oplus ...\oplus V_r$ where each $V_i$ is $f$-invariant, $\mathrm{dim}V_i=n_i$ and, when restricted to $V_i$, $f$ has the form $\lambda_iI+N_i$ where $I$ is the identity mapping on $V_i$ and each $N_i$ is nilpotent (see e.g. [a1]).

An equivalent and perhaps more familiar description of $f$ is through the matrix $A$ representing $f$ with respect to a particular basis of $V$. The same theory says that this basis may be chosen so that $A$ takes its Jordan canonical form

\begin{equation}A=\begin{pmatrix}A_1 & \Box & \Box \\ \Box & \ddots & \Box \\ \Box & \Box & A_r\end{pmatrix}.\end{equation}

Here, each $A_i$ is an $(n_i\times n_i)$-matrix with $\lambda_i$ in each diagonal position and an entry $1$ or $0$ in each superdiagonal position (and it is assumed that each unnamed entry in any matrix is zero). Each matrix $A_i$ can then be written in the form

\begin{equation}A_i=\begin{pmatrix}B_{i_{1}}& \Box & \Box \\ \Box & \ddots & \Box \\ \Box & \Box & B_{i_{k(i)}}\end{pmatrix},\end{equation}

where each $B_{i_{j}}$ is an $(m_{i_{j}}\times m_{i_{j}})$-matrix and $m_{i_{1}}\geq ...\geq m_{i_{k(i)}}$ with $\sum_jm_{i_{j}}=n_i$.

A shorthand way of indicating this structure for $f$ is by means of its Segre symbol (or Segre characteristic), which is

\begin{equation}\begin{Bmatrix}(m_{1_{1}}...m_{1_{k(1)}})...(m_{r_{1}}...m_{r_{k(r)}})\end{Bmatrix}.\end{equation}

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 $F=C$. For vector spaces over $\textbf{R}$ the above does not apply, since $\textbf{R}$ 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 $\textbf{R}$. 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 $z\overline{z}$.

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 $4$-dimensional connected Hausdorff manifold admitting a Lorentz metric $g$ 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 $p$ in $M$ by regarding them as linear mappings on the tangent space $T_p(M)$ to $M$ at $p$. Since $T_p(M)$ is a real vector space one, of course, encounters the problem of closure mentioned above. However, if the tensor in question is $S$, then one has the eigenvector-eigenvalue problem

\begin{equation}S^{\ddot{a}}\Box_bk^b=\lambda k^{\ddot{a}}g_{\ddot{a}c}S^c\Box_b\mathrm{symmetric},\end{equation}

for a complex eigenvector and eigenvalue $k$ and $\lambda$, respectively. It should be noted here that one could write (4) as $S_{\ddot{a}b}k^b=\lambda g_{\ddot{a}b}k^b$, but because of the Lorentz signature of $g$ 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 (4) 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 (4) and the signature of $g$, are $\{1111\}$, $\{211\}$, $\{31\}$ and $\{z\overline{z}11\}$ 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)