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Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120070/s1200701.png" /> be an <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120070/s1200702.png" />-dimensional [[Vector space|vector space]] over an [[Algebraically closed field|algebraically closed field]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120070/s1200703.png" /> (say <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120070/s1200704.png" />, but not <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120070/s1200705.png" />; this special case will be dealt with separately) and let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120070/s1200706.png" /> be a [[Linear operator|linear operator]] from <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120070/s1200707.png" /> to itself. The essential information in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120070/s1200708.png" /> can be described by the theory of the Jordan canonical form (cf. [[Normal form|Normal form]]). The essential result is that if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120070/s1200709.png" /> are the distinct eigenvalues of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120070/s12007010.png" /> with multiplicities <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120070/s12007011.png" />, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120070/s12007012.png" /> is the direct sum <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120070/s12007013.png" /> where each <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120070/s12007014.png" /> is <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120070/s12007015.png" />-invariant, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120070/s12007016.png" /> and, when restricted to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120070/s12007017.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120070/s12007018.png" /> has the form <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120070/s12007019.png" /> where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120070/s12007020.png" /> is the identity mapping on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120070/s12007021.png" /> and each <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120070/s12007022.png" /> is nilpotent (see e.g. [[#References|[a1]]]).
+
Let $V$ be an $n$-dimensional [[vector space]] over an [[algebraically closed field]] $F$ (say $F=C$, but not $\mathbf{R}$; this special case will be dealt with separately) and let $f$ be a [[Linear operator|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|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. [[#References|[a1]]]).
  
An equivalent and perhaps more familiar description of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120070/s12007023.png" /> is through the [[Matrix|matrix]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120070/s12007024.png" /> representing <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120070/s12007025.png" /> with respect to a particular basis of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120070/s12007026.png" />. The same theory says that this basis may be chosen so that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120070/s12007027.png" /> takes its Jordan canonical form
+
An equivalent and perhaps more familiar description of $f$ is through the [[Matrix|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
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120070/s12007028.png" /></td> </tr></table>
+
\begin{equation}A=\begin{pmatrix}A_1 & {} & {} \\ {} & \ddots & {} \\ {} & {} & A_r\end{pmatrix}.\end{equation}
  
Here, each <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120070/s12007029.png" /> is an <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120070/s12007030.png" />-matrix with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120070/s12007031.png" /> in each diagonal position and an entry <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120070/s12007032.png" /> or <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120070/s12007033.png" /> in each superdiagonal position (and it is assumed that each unnamed entry in any matrix is zero). Each matrix <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120070/s12007034.png" /> can then be written in the form
+
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
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120070/s12007035.png" /></td> </tr></table>
+
\begin{equation}A_i=\begin{pmatrix}B_{i_{1}}& {} & {} \\ {} & \ddots & {} \\ {} & {} & B_{i_{k(i)}}\end{pmatrix},\end{equation}
  
where each <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120070/s12007036.png" /> is an <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120070/s12007037.png" />-matrix and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120070/s12007038.png" /> with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120070/s12007039.png" />.
+
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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120070/s12007040.png" /> is by means of its Segre symbol (or Segre characteristic), which is
+
A shorthand way of indicating this structure for $f$ is by means of its Segre symbol (or [[Segre characteristic of a square matrix|Segre characteristic]]), which is
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120070/s12007041.png" /></td> </tr></table>
+
\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.
 
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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120070/s12007042.png" />. For vector spaces over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120070/s12007043.png" /> the above does not apply, since <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120070/s12007044.png" /> is not algebraically closed and the proof relies on the fundamental theorem of algebra (cf. also [[Algebra, fundamental theorem of|Algebra, fundamental theorem of]]). Of course, the Jordan forms still apply if the characteristic equation can be solved completely over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120070/s12007045.png" />. If not, an alternative approach (the rational canonical form) is available [[#References|[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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120070/s12007046.png" />.
+
An obvious important application of the theory is when $F=C$. For vector spaces over $\mathbf{R}$ the above does not apply, since $\mathbf{R}$ is not algebraically closed and the proof relies on the fundamental theorem of algebra (cf. also [[Algebra, fundamental theorem of|Algebra, fundamental theorem of]]). Of course, the Jordan forms still apply if the characteristic equation can be solved completely over $\mathbf{R}$. If not, an alternative approach (the rational canonical form) is available [[#References|[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|Petrov classification]] of gravitational fields [[#References|[a3]]]. Another also appears in Einstein's theory and to briefly describe it the following definitions are required.
 
One of the important modern applications of the Segre classification is the [[Petrov classification|Petrov classification]] of gravitational fields [[#References|[a3]]]. Another also appears in Einstein's theory and to briefly describe it the following definitions are required.
  
A [[Space-time|space-time]] is a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120070/s12007047.png" />-dimensional connected Hausdorff [[Manifold|manifold]] admitting a Lorentz metric <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120070/s12007048.png" /> of signature <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120070/s12007049.png" />. 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|Tensor on a vector space]]). Following the success of the Petrov classification it has been found useful to classify such tensors at a point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120070/s12007050.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120070/s12007051.png" /> by regarding them as linear mappings on the tangent space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120070/s12007052.png" /> to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120070/s12007053.png" /> at <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120070/s12007054.png" />. Since <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120070/s12007055.png" /> is a real vector space one, of course, encounters the problem of closure mentioned above. However, if the tensor in question is <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120070/s12007056.png" />, then one has the eigenvector-eigenvalue problem
+
A [[Space-time|space-time]] is a $4$-dimensional connected Hausdorff [[Manifold|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|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
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120070/s12007057.png" /></td> <td valign="top" style="width:5%;text-align:right;">(a1)</td></tr></table>
+
\begin{equation}S^{a}{}_b k^b=\lambda k^{a}, \quad g_{ac}S^c{}_b \text{ symmetric},\end{equation}
  
for a complex eigenvector and eigenvalue <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120070/s12007058.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120070/s12007059.png" />, respectively. It should be noted here that one could write (a1) as <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120070/s12007060.png" />, but because of the Lorentz signature of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120070/s12007061.png" /> 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 [[#References|[a4]]], [[#References|[a3]]], [[#References|[a5]]], [[#References|[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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120070/s12007062.png" />, are <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120070/s12007063.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120070/s12007064.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120070/s12007065.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120070/s12007066.png" /> 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|relativity theory]] (see e.g. [[#References|[a7]]]).
+
for a complex eigenvector and eigenvalue $k$ and $\lambda$, respectively. It should be noted here that one could write (4) as $S_{ab}k^b=\lambda g_{ab}k^b$, 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 [[#References|[a4]]], [[#References|[a3]]], [[#References|[a5]]], [[#References|[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. [[#References|[a7]]]).
  
 
====References====
 
====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top"> D.T. Finkbeiner,   "Introduction to matrices and linear transformations" , Freeman  (1960)</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top"> G. Birkhoff,   S. MacLane,   "A survey of modern algebra" , Macmillan  (1961)</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top"> A.Z. Petrov,   "Einstein spaces" , Pergamon  (1969)</TD></TR><TR><TD valign="top">[a4]</TD> <TD valign="top"> S.W. Hawking,   G.F.R. Ellis,   "The large scale structure of space-time" , Cambridge Univ. Press  (1973)</TD></TR><TR><TD valign="top">[a5]</TD> <TD valign="top"> J.F. Plebanski,   "The algebraic structure of the tensor of matter"  ''Acta Phys. Polon.'' , '''26'''  (1964)  pp. 963</TD></TR><TR><TD valign="top">[a6]</TD> <TD valign="top"> G.S. Hall,   "Differential geometry" , ''Banach Centre Publ.'' , '''12''' , Banach Centre  (1984)  pp. 53</TD></TR><TR><TD valign="top">[a7]</TD> <TD valign="top"> D. Kramer,   H. Stephani,   M.A.H. MacCallum,   E. Herlt,   "Exact solutions of Einstein's field equations" , Cambridge Univ. Press  (1980)</TD></TR></table>
+
<table>
 +
<TR><TD valign="top">[a1]</TD> <TD valign="top"> D.T. Finkbeiner, "Introduction to matrices and linear transformations" , Freeman  (1960)</TD></TR>
 +
<TR><TD valign="top">[a2]</TD> <TD valign="top"> G. Birkhoff, S. MacLane, "A survey of modern algebra" , Macmillan  (1961)</TD></TR>
 +
<TR><TD valign="top">[a3]</TD> <TD valign="top"> A.Z. Petrov, "Einstein spaces" , Pergamon  (1969)</TD></TR>
 +
<TR><TD valign="top">[a4]</TD> <TD valign="top"> S.W. Hawking, G.F.R. Ellis, "The large scale structure of space-time" , Cambridge Univ. Press  (1973)</TD></TR>
 +
<TR><TD valign="top">[a5]</TD> <TD valign="top"> J.F. Plebanski, "The algebraic structure of the tensor of matter"  ''Acta Phys. Polon.'' , '''26'''  (1964)  pp. 963</TD></TR>
 +
<TR><TD valign="top">[a6]</TD> <TD valign="top"> G.S. Hall, "Differential geometry" , ''Banach Centre Publ.'' , '''12''' , Banach Centre  (1984)  pp. 53</TD></TR>
 +
<TR><TD valign="top">[a7]</TD> <TD valign="top"> D. Kramer, H. Stephani, M.A.H. MacCallum, E. Herlt, "Exact solutions of Einstein's field equations" , Cambridge Univ. Press  (1980)</TD></TR>
 +
</table>

Latest revision as of 06:48, 16 March 2024

Let $V$ be an $n$-dimensional vector space over an algebraically closed field $F$ (say $F=C$, but not $\mathbf{R}$; 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 & {} & {} \\ {} & \ddots & {} \\ {} & {} & 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}}& {} & {} \\ {} & \ddots & {} \\ {} & {} & 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 $\mathbf{R}$ the above does not apply, since $\mathbf{R}$ 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 $\mathbf{R}$. 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^{a}{}_b k^b=\lambda k^{a}, \quad g_{ac}S^c{}_b \text{ symmetric},\end{equation}

for a complex eigenvector and eigenvalue $k$ and $\lambda$, respectively. It should be noted here that one could write (4) as $S_{ab}k^b=\lambda g_{ab}k^b$, 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)
How to Cite This Entry:
Segre classification. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Segre_classification&oldid=18578
This article was adapted from an original article by G.S. Hall (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article