Signature (linear algebra)

The signature (also the index of inertia or index ) is an object from mathematics , which is mainly considered in linear algebra but also in different areas of differential geometry . It is precisely a triple of numbers that is an invariant of a symmetrical bilinear form . This triple number is therefore in particular independent of the basic choice with respect to which the bilinear form is represented. Sylvester's theorem of inertia , named after the mathematician James Joseph Sylvester, is fundamental to the definition of the signature . Therefore the signature is sometimes called the New Year's signature .

definition

Let be a finite-dimensional real vector space and a symmetric bilinear form with the representation matrix ${\displaystyle V}$ ${\displaystyle s\colon V\times V\rightarrow \mathbb {R} }$

${\displaystyle A:={\begin{pmatrix}1&0&0&0&\ldots &0&0&\ldots &0\\0&\ddots &0&&&&&&\vdots \\0&0&1&0&&&&&0\\0&&0&-1&0&&&&0\\\vdots &&&0&\ddots &0&&&\vdots \\0&&&&0&-1&0&&0\\0&&&&&0&0&0&0\\\vdots &&&&&&0&\ddots &0\\0&\ldots &0&0&\ldots &0&0&0&0\end{pmatrix}}}$.

This matrix has the entries , and , on the main diagonal , all other coefficients are . ${\displaystyle 1}$ ${\displaystyle -1}$ ${\displaystyle 0}$ ${\displaystyle 0}$

With is now the number of entries, with the number of entries and with the number of entries. Then it's called triple ${\displaystyle r_{+}(s)}$ ${\displaystyle 1}$ ${\displaystyle r_{-}(s)}$ ${\displaystyle -1}$ ${\displaystyle r_{0}(s)}$ ${\displaystyle 0}$

${\displaystyle \sigma (s):=(r_{+}(s),r_{-}(s),r_{0}(s))}$

Inertia index or (New Year's) signature of . Since, according to Sylvester's law of inertia, every symmetrical bilinear form has a diagonal matrix like a representation matrix , the signature for all symmetrical bilinear forms is well defined . ${\displaystyle s}$ ${\displaystyle A}$

If there are no zero entries on the main diagonal of the display matrix ( i.e. if the symmetrical bilinear form has not degenerated ), the coefficient is sometimes omitted and this is called a tuple ${\displaystyle A}$ ${\displaystyle r_{0}(s)}$

${\displaystyle \sigma (s):=(r_{+}(s),r_{-}(s))}$

the signature of . Occasionally, too ${\displaystyle s}$

${\displaystyle \operatorname {sign} (s):=r_{+}(s)-r_{-}(s)}$

referred to as a signature (especially if there is no degeneration). Sometimes it is also called an index . ${\displaystyle r_{-}(s)}$

The term signature is also used for symmetric matrices . It then designates the signature of the symmetrical bilinear form defined by for . ${\displaystyle A\in \mathbb {R} ^{n\times n}}$ ${\displaystyle s(x,y)=x^{T}Ay}$ ${\displaystyle x,y\in \mathbb {R} ^{n}}$

Signature of the Minkowski metric

An important example from physics is the Minkowski metric of the special theory of relativity . This is a symmetrical bilinear form with the representation matrix

${\displaystyle \eta =\pm {\begin{pmatrix}1&0&0&0\\0&-1&0&0\\0&0&-1&0\\0&0&0&-1\end{pmatrix}}}$.

The entry in the top left of the matrix stands for the time coordinate, which has the opposite sign to the other three spatial coordinates. The signature in which the time has a positive sign is also called the West Coast convention , written and in English-language literature . The reverse signature as written and called the East Coast convention . ${\displaystyle \eta _{00}}$ ${\displaystyle (1,3)}$ ${\displaystyle (+,-,-,-)}$ ${\displaystyle (3,1)}$ ${\displaystyle (-,+,+,+)}$

Using the signature of the metric, a vector can be classified as time-like, light-like or space-like based on its scalar product . The following applies to the East Coast convention : ${\displaystyle u}$ ${\displaystyle \eta (u,u)}$ ${\displaystyle (-,+,+,+)}$

• ${\displaystyle \eta (u,u)>0}$ space-like
• ${\displaystyle \eta (u,u)=0}$ light-like
• ${\displaystyle \eta (u,u)<0}$ timely

and for the West Coast convention : ${\displaystyle (+,-,-,-)}$

• ${\displaystyle \eta (u,u)>0}$ timely
• ${\displaystyle \eta (u,u)=0}$ light-like
• ${\displaystyle \eta (u,u)<0}$ space-like

Algorithm for determining the signature

In order to calculate the signature of a symmetrical bilinear form, it is not necessary to determine the base change of the representation matrix from . After any representation matrix (not necessarily in diagonal form) of the symmetrical bilinear form has been determined, it can also be understood as a representation matrix of an endomorphism . The eigenvalues can then be determined from this matrix . If one then denotes with the number of positive eigenvalues, with the number of negative eigenvalues ​​and with the multiplicity of the eigenvalues , then corresponds to ${\displaystyle s\colon V\times V\rightarrow \mathbb {R} }$ ${\displaystyle s}$ ${\displaystyle B}$ ${\displaystyle s}$ ${\displaystyle r_{+}(s)}$ ${\displaystyle r_{-}(s)}$ ${\displaystyle r_{0}(s)}$ ${\displaystyle 0}$

${\displaystyle \sigma (s):=(r_{+}(s),r_{-}(s),r_{0}(s))}$

the signature of . ${\displaystyle s}$

example

Let be a symmetric bilinear form. Thus, the matrix representing which has canonical base form ${\displaystyle s(x,y)={\tfrac {1}{2}}x_{1}y_{2}+{\tfrac {1}{2}}y_{1}x_{2}}$

${\displaystyle M_{\mathcal {K}}(s)={\begin{pmatrix}0&{\frac {1}{2}}\\{\frac {1}{2}}&0\end{pmatrix}}}$.

If one understands this matrix in the meantime as a self adjoint endomorphism of , one knows from the spectral theorem that there is an orthonormal basis from eigenvectors , so that it has a diagonal shape. If you multiply each eigenvector by , where is the corresponding eigenvalue, and then carry out the basic transformation, you get a diagonal matrix with entries 1 and −1 on the diagonal. Here you can read the signature directly. In our specific example the eigenvalues ​​are and and the orthonormal eigenvectors are and . If you multiply this basis as described above , you get a transformation matrix ${\displaystyle \mathbb {R} ^{2}}$ ${\displaystyle S^{t}M_{\mathcal {K}}(s)S}$ ${\displaystyle |\lambda _{i}|^{-{\frac {1}{2}}}}$ ${\displaystyle \lambda _{i}}$ ${\displaystyle {\tfrac {1}{2}}}$ ${\displaystyle -{\tfrac {1}{2}}}$ ${\displaystyle \textstyle {\tfrac {1}{\sqrt {2}}}\left({\begin{smallmatrix}-1\\1\end{smallmatrix}}\right)}$ ${\displaystyle {\tfrac {1}{\sqrt {2}}}\left({\begin{smallmatrix}1\\1\end{smallmatrix}}\right)}$ ${\displaystyle |\lambda _{i}|^{-{\frac {1}{2}}}}$

${\displaystyle T={\begin{pmatrix}-1&1\\1&1\end{pmatrix}}}$

and the base transformation looks like this:

${\displaystyle T^{t}M_{\mathcal {K}}(s)T={\begin{pmatrix}-1&1\\1&1\end{pmatrix}}{\begin{pmatrix}0&{\frac {1}{2}}\\{\frac {1}{2}}&0\end{pmatrix}}{\begin{pmatrix}-1&1\\1&1\end{pmatrix}}={\begin{pmatrix}-1&0\\0&1\end{pmatrix}}}$

So the bilinear form assigned to the matrix has the signature . In this example, however, one must note that bilinear forms have no eigenvalues ​​and that the route via the eigenvalues ​​is just a trick for computing. ${\displaystyle (1,1,0)}$

The above diagonal shape could also be calculated using the Gaussian algorithm , in which transformations are always applied equally to rows and columns.

Special case

A symmetrical, non-singular matrix is ​​given. Then the signature is given by:

${\displaystyle \mathrm {sign} (A)=\mathrm {sgn} (A_{1})+v_{g}-v_{a}\;.}$

Here refers to the first principal minor of . The other two quantities result from the calculation of the determinants of the other minors, whereby only the sign is important. is the number of constant signs from to and the number of sign changes from to . ${\displaystyle A_{1}}$ ${\displaystyle A}$ ${\displaystyle v_{g}}$ ${\displaystyle \det(A_{k})}$ ${\displaystyle \det(A_{k+1})}$ ${\displaystyle v_{a}}$ ${\displaystyle \det(A_{k})}$ ${\displaystyle \det(A_{k+1})}$

The signature in the differential geometry

Signature of a pseudo-Riemannian manifold

→ Main article : Pseudo-Riemannian manifold

In differential geometry , symmetrical bilinear forms are generalized to differentiable manifolds in the form of symmetrical covariant smooth tensor fields of the second order. Such a tensor field then acts as a bilinear form at every point on the respective tangent space . If the signature of the respective bilinear form is the same at every point of the manifold and if these are not degenerate, one speaks of a pseudo-Riemannian metric and calls a manifold that is provided with such a metric a pseudo-Riemannian manifold . Such manifolds are the subject of investigation in pseudo-Riemannian geometry and play an important role in physics.

Signature of a manifold

In global analysis , a sub-area of ​​differential geometry, one considers the signature of a manifold. In order to define the signature of such a "curved space", a special bilinear form is chosen and it is determined that its signature is the signature of the manifold. The signature operator is a key message in this context. It relates the signature, which is an invariant of the bilinear form, to an invariant of the manifold.

Let be a compact , orientable smooth manifold whose dimension is divisible by . In addition, the De Rham cohomology of is denoted by. Consider the bilinear shape passed by ${\displaystyle M}$ ${\displaystyle n}$ ${\displaystyle 4}$ ${\displaystyle H^{*}(M)}$ ${\displaystyle M}$ ${\displaystyle s\colon H^{\frac {n}{2}}(M)\times H^{\frac {n}{2}}(M)\to \mathbb {R} }$

${\displaystyle (\alpha ,\beta )\mapsto \int _{M}\alpha \wedge \beta }$

is defined. This is symmetrical and non-degenerate due to the Poincaré duality , that is . Then the signature of the manifold is defined as the signature of the bilinear form , that is ${\displaystyle r_{0}(s)=0}$ ${\displaystyle \operatorname {sign} (M)}$ ${\displaystyle M}$ ${\displaystyle \operatorname {sign} (s)}$ ${\displaystyle s}$

${\displaystyle \operatorname {sign} (M):=\operatorname {sign} (s)=r_{+}(s)-r_{-}(s).}$

literature

• Gerd Fischer: Linear Algebra. Vieweg-Verlag, Wiesbaden 2003, ISBN 3-528-03217-0 .
• R. Abraham, JE Marsden, T. Ratiu: Manifolds, Tensor Analysis, and Applications . Springer , Berlin 2003, ISBN 3-540-96790-7 .

Individual evidence

1. ^ Abraham, Marsden, Ratiu, p. 398.
2. ↑ Craig Callender: What Makes Time Special? Oxford University Press, 2017, ISBN 978-0-19-879730-2 , pp. 123 ( limited preview in Google Book search).
3. ^ Nicole Berlin, Ezra Getzler , Michèle Vergne : Heat Kernels and Dirac Operators . Springer-Verlag, Berlin a. a. 2004, ISBN 3-540-20062-2 , pp. 128-129.