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Tensor[QuadraticFormSignature] - find the signature of a covariant, symmetric, rank 2 tensor

Calling Sequences

QuadraticFormSignature(Q, B, option)

Parameters

Q       - a covariant, symmetric, rank 2 tensor, possibly degenerate

B       - (optional) a list of vectors spanning a subspace of the vector space upon which the tensor Qis defined

option  - keyword argument output = "dimensions"

Description

 • Let be a vector space and let  be a covariant, symmetric, rank 2 tensor (quadratic form) defined on The null space of is defined to be for allThe vector space  decomposes into a direct sum

where is positive-definite on and negative-definite on that is, for alland for allEquality holds if and only if . The subspace is unique but the subspaces and are not unique. Nevertheless, Sylvester's law of inertia states that the dimensions of and are uniquely determined by $Q$.

 • The quadratic form is called non-degenerate if 0. In this case the dimensions of and specify the signature of
 • The command returns either a list of lists of vectors spanning the subspaces or a list of their dimensions.
 • The algorithm that is used is as follows. Let be a basis for $V$. First, calculate the null space $N$ of Use the command ComplementaryBasis to find vectors such that

Search through the set of vectors {to find a vector Let be the orthogonal complement of by $Q.$ Write If   then or  Continue in this way to arrive at (*). Note that this method does not require the calculation of the eigenvalues/eigenvectors of $Q.$

 • If the quadratic form depends upon parameters, then the assuming facility may be useful in determining the sign of at each step in the algorithm.

Examples

 > $\mathrm{with}\left(\mathrm{DifferentialGeometry}\right):$$\mathrm{with}\left(\mathrm{Tensor}\right):$

Example 1.

Find the signature of 4 different quadratic forms defined on the tangent space at a point of a 4-dimensional manifold.

 > $\mathrm{DGsetup}\left(\left[\mathrm{x1},\mathrm{x2},\mathrm{x3},\mathrm{x4}\right],M\right)$
 ${\mathrm{frame name: M}}$ (2.1)

First quadratic form.

 > $\mathrm{g1}≔\mathrm{evalDG}\left(\mathrm{dx1}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}&t\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}\mathrm{dx1}+\mathrm{dx2}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}&t\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}\mathrm{dx2}+\mathrm{dx3}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}&t\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}\mathrm{dx3}+\mathrm{dx4}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}&t\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}\mathrm{dx4}\right)$
 ${\mathrm{g1}}{:=}{\mathrm{dx1}}{}{\mathrm{dx1}}{+}{\mathrm{dx2}}{}{\mathrm{dx2}}{+}{\mathrm{dx3}}{}{\mathrm{dx3}}{+}{\mathrm{dx4}}{}{\mathrm{dx4}}$ (2.2)
 > $\mathrm{QuadraticFormSignature}\left(\mathrm{g1}\right)$
 $\left[\left[{\mathrm{D_x1}}{,}{\mathrm{D_x2}}{,}{\mathrm{D_x3}}{,}{\mathrm{D_x4}}\right]{,}\left[{}\right]{,}\left[{}\right]\right]$ (2.3)

We see that the quadratic form is positive-definite in all directions; it is a Riemannian metric.

Second quadratic form.

 M > $\mathrm{g2}≔\mathrm{evalDG}\left(-\mathrm{dx1}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}&t\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}\mathrm{dx1}+\mathrm{dx2}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}&t\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}\mathrm{dx2}+\mathrm{dx3}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}&t\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}\mathrm{dx3}+\mathrm{dx4}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}&t\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}\mathrm{dx4}\right)$
 ${\mathrm{g2}}{:=}{-}\left({\mathrm{dx1}}{}{\mathrm{dx1}}\right){+}{\mathrm{dx2}}{}{\mathrm{dx2}}{+}{\mathrm{dx3}}{}{\mathrm{dx3}}{+}{\mathrm{dx4}}{}{\mathrm{dx4}}$ (2.4)
 M > $\mathrm{QuadraticFormSignature}\left(\mathrm{g2}\right)$
 $\left[\left[{\mathrm{D_x2}}{,}{\mathrm{D_x3}}{,}{\mathrm{D_x4}}\right]{,}\left[{\mathrm{D_x1}}\right]{,}\left[{}\right]\right]$ (2.5)

The quadratic form is positive-definite in the 3 directions [ and negative-definite in the 1 direction ${\mathrm{D}}_{\mathrm{x1}}$; it is a Lorentzian metric.



Third quadratic form.

 M > $\mathrm{g3}≔\mathrm{evalDG}\left(2\mathrm{dx1}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}&s\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}\mathrm{dx2}+2\mathrm{dx3}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}&s\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}\mathrm{dx4}\right)$
 ${\mathrm{g3}}{:=}{\mathrm{dx1}}{}{\mathrm{dx2}}{+}{\mathrm{dx2}}{}{\mathrm{dx1}}{+}{\mathrm{dx3}}{}{\mathrm{dx4}}{+}{\mathrm{dx4}}{}{\mathrm{dx3}}$ (2.6)
 M > $\mathrm{QuadraticFormSignature}\left(\mathrm{g3}\right)$
 $\left[\left[{\mathrm{D_x1}}{+}{\mathrm{D_x2}}{,}{\mathrm{D_x3}}{+}{\mathrm{D_x4}}\right]{,}\left[{\mathrm{D_x1}}{-}{\mathrm{D_x2}}{,}{\mathrm{D_x3}}{-}{\mathrm{D_x4}}\right]{,}\left[{}\right]\right]$ (2.7)

The quadratic form is positive-definite in the 2 directions [ and negative-definite in the 2 directions [- .

Fourth quadratic form.

 M > $\mathrm{g4}≔\mathrm{evalDG}\left(\mathrm{dx1}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}&t\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}\mathrm{dx1}+\mathrm{dx2}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}&t\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}\mathrm{dx2}-\mathrm{dx3}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}&t\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}\mathrm{dx3}\right)$
 ${\mathrm{g4}}{:=}{\mathrm{dx1}}{}{\mathrm{dx1}}{+}{\mathrm{dx2}}{}{\mathrm{dx2}}{-}\left({\mathrm{dx3}}{}{\mathrm{dx3}}\right)$ (2.8)
 M > $\mathrm{QuadraticFormSignature}\left(\mathrm{g4}\right)$
 $\left[\left[{\mathrm{D_x1}}{,}{\mathrm{D_x2}}\right]{,}\left[{\mathrm{D_x3}}\right]{,}\left[{\mathrm{D_x4}}\right]\right]$ (2.9)

The quadratic form is positive-definite in the 2 directions [ and negative-definite in the direction [${\mathrm{D}}_{\mathrm{x3}}]$ and degenerate in the direction [${\mathrm{D}}_{\mathrm{x4}}]$. Here are the dimensions of these spaces.

 M > $\mathrm{QuadraticFormSignature}\left(\mathrm{g4},\mathrm{output}="dimensions"\right)$
 $\left[{2}{,}{1}{,}{1}\right]$ (2.10)

Example 2.

We calculate the signature of the quadratic forms restricted to some subspaces.

 M > $\mathrm{QuadraticFormSignature}\left(\mathrm{g2},\left[\mathrm{D_x1},\mathrm{D_x2},\mathrm{D_x3}\right]\right)$
 $\left[\left[{\mathrm{D_x2}}{,}{\mathrm{D_x3}}\right]{,}\left[{\mathrm{D_x1}}\right]{,}\left[{}\right]\right]$ (2.11)
 M > $\mathrm{QuadraticFormSignature}\left(\mathrm{g4},\left[\mathrm{D_x1},\mathrm{D_x3},\mathrm{D_x4}\right]\right)$
 $\left[\left[{\mathrm{D_x1}}\right]{,}\left[{\mathrm{D_x3}}\right]{,}\left[{\mathrm{D_x4}}\right]\right]$ (2.12)

Example 3.

Here we consider quadratic forms which depend upon a parameter.

 M > $\mathrm{DGsetup}\left(\left[\mathrm{x1},\mathrm{x2},\mathrm{x3},\mathrm{x4}\right],M\right)$
 ${\mathrm{frame name: M}}$ (2.13)
 > $\mathrm{g5}≔\mathrm{evalDG}\left(a\mathrm{dx1}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}&tensor\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}\mathrm{dx1}-\mathrm{dx2}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}&t\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}\mathrm{dx2}+\mathrm{dx3}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}&t\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}\mathrm{dx3}+\mathrm{dx4}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}&t\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}\mathrm{dx4}\right)$
 ${\mathrm{g5}}{:=}\left({a}{}{\mathrm{dx1}}\right){}{\mathrm{dx1}}{-}\left({\mathrm{dx2}}{}{\mathrm{dx2}}\right){+}{\mathrm{dx3}}{}{\mathrm{dx3}}{+}{\mathrm{dx4}}{}{\mathrm{dx4}}$ (2.14)

Without further information on a it is not possible to compute the signature, that changes depending on the sign of $a:$

 M > $\mathrm{QuadraticFormSignature}\left(\mathrm{g5}\right)$
 ${\mathrm{FAIL}}$ (2.15)
 M > $\mathrm{QuadraticFormSignature}\left(\mathrm{g5}\right)\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}\mathbf{assuming}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}0
 $\left[\left[{\mathrm{D_x1}}{,}{\mathrm{D_x3}}{,}{\mathrm{D_x4}}\right]{,}\left[{\mathrm{D_x2}}\right]{,}\left[{}\right]\right]$ (2.16)
 M > $\mathrm{QuadraticFormSignature}\left(\mathrm{g5}\right)\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}\mathbf{assuming}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}a<0$
 $\left[\left[{\mathrm{D_x3}}{,}{\mathrm{D_x4}}\right]{,}\left[{\mathrm{D_x1}}{,}{\mathrm{D_x2}}\right]{,}\left[{}\right]\right]$ (2.17)

For more complicated examples, use infolevel to trace the testing performed by the QuadraticFormSignature procedure to see exactly at what point in the algorithm the procedure returns $\mathrm{FAIL}$.

 M > $\mathrm{g6}≔\mathrm{evalDG}\left(a\mathrm{dx1}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}&tensor\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}\mathrm{dx1}+2\mathrm{dx1}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}&s\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}\mathrm{dx2}-\mathrm{dx2}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}&t\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}\mathrm{dx2}+\mathrm{dx3}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}&t\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}\mathrm{dx3}+\mathrm{dx4}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}&t\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}\mathrm{dx4}\right)$
 ${\mathrm{g6}}{:=}\left({a}{}{\mathrm{dx1}}\right){}{\mathrm{dx1}}{+}{\mathrm{dx1}}{}{\mathrm{dx2}}{+}{\mathrm{dx2}}{}{\mathrm{dx1}}{-}\left({\mathrm{dx2}}{}{\mathrm{dx2}}\right){+}{\mathrm{dx3}}{}{\mathrm{dx3}}{+}{\mathrm{dx4}}{}{\mathrm{dx4}}$ (2.18)
 M > $\mathrm{QuadraticFormSignature}\left(\mathrm{g6}\right)$
 ${\mathrm{FAIL}}$ (2.19)

Now set the infolevel to 2.

 M > $\mathrm{infolevel}\left[\mathrm{QuadraticFormSignature}\right]≔2$
 ${{\mathrm{infolevel}}}_{{\mathrm{DifferentialGeometry:-Tensor:-QuadraticFormSignature}}}{:=}{2}$ (2.20)
 M > $\mathrm{QuadraticFormSignature}\left(\mathrm{g6}\right)$
 The null space of the metric is    Testing vector: D_x1       The norm of this vector is: a    Testing vector: D_x2       The norm of this vector is: -1    Testing vector: D_x1+D_x2       The norm of this vector is: a+1    Testing vector: D_x3       The norm of this vector is: 1    Testing vector: D_x1+D_x2       The norm of this vector is: a+1    Testing vector: D_x4       The norm of this vector is: 1    Testing vector: D_x1+D_x2       The norm of this vector is: a+1
 ${\mathrm{FAIL}}$ (2.21)

We see that the signature depends on the sign of $a+1$

 M > $\mathrm{infolevel}\left[\mathrm{QuadraticFormSignature}\right]≔0$
 ${{\mathrm{infolevel}}}_{{\mathrm{DifferentialGeometry:-Tensor:-QuadraticFormSignature}}}{:=}{0}$ (2.22)
 M > $\mathrm{QuadraticFormSignature}\left(\mathrm{g6}\right)\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}\mathbf{assuming}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}a<-1$
 $\left[\left[{\mathrm{D_x3}}{,}{\mathrm{D_x4}}\right]{,}\left[{\mathrm{D_x1}}{,}{\mathrm{D_x1}}{-}\left({a}{}{\mathrm{D_x2}}\right)\right]{,}\left[{}\right]\right]$ (2.23)
 M > $\mathrm{QuadraticFormSignature}\left(\mathrm{g6}\right)\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}\mathbf{assuming}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}-1
 $\left[\left[{\mathrm{D_x1}}{+}{\mathrm{D_x2}}{,}{\mathrm{D_x3}}{,}{\mathrm{D_x4}}\right]{,}\left[{\mathrm{D_x2}}\right]{,}\left[{}\right]\right]$ (2.24)

 See Also