Let $V$be a vector space and let $Q$ be a covariant, symmetric, rank 2 tensor (quadratic form) defined on $V\.$The null space of $Q$is defined to be $N \= \{ X\in V \| Q\left(Y\,X\right) \= 0$for all$Y \in V\}\.$The vector space $V$ decomposes into a direct sum

The quadratic form $Q$is called non-degenerate if $N \=$0. In this case the dimensions of $V_{\+}$and $V_{-}$specify the signature of $Q\.$

The command $\mathrm{QuadraticFormSignature}$returns either a list of lists of vectors spanning the subspaces $V_{\+} \,V{ }_{-}\, N$or a list of their dimensions.

 The algorithm that is used is as follows. Let $B$be a basis for $V$. First, calculate the null space $N$ of $Q \.$Use the command ComplementaryBasis to find vectors $X_i \in B$such that

If the quadratic form depends upon parameters, then the assuming facility may be useful in determining the sign of $Q\left(Y_1\, Y_1\right)$at each step in the algorithm.

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

$\mathrm{DGsetup}\left(\left[\mathrm{x1}\,\mathrm{x2}\,\mathrm{x3}\,\mathrm{x4}\right]\,M\right)$

$\mathrm{frame\; name:\; M}$

$\mathrm{g1}≔\mathrm{evalDG}\left(\mathrm{dx1}\&t\mathrm{dx1}+\mathrm{dx2}\&t\mathrm{dx2}+\mathrm{dx3}\&t\mathrm{dx3}+\mathrm{dx4}\&t\mathrm{dx4}\right)$

$\mathrm{g1}:=\mathrm{dx1}\mathrm{dx1}+\mathrm{dx2}\mathrm{dx2}+\mathrm{dx3}\mathrm{dx3}+\mathrm{dx4}\mathrm{dx4}$

$\mathrm{QuadraticFormSignature}\left(\mathrm{g1}\right)$

$\left[\left[\mathrm{D\_x1}\,\mathrm{D\_x2}\,\mathrm{D\_x3}\,\mathrm{D\_x4}\right]\,\left[\right]\,\left[\right]\right]$

$\mathrm{g2}≔\mathrm{evalDG}\left(-\mathrm{dx1}\&t\mathrm{dx1}+\mathrm{dx2}\&t\mathrm{dx2}+\mathrm{dx3}\&t\mathrm{dx3}+\mathrm{dx4}\&t\mathrm{dx4}\right)$

$\mathrm{g2}:=-\left(\mathrm{dx1}\mathrm{dx1}\right)+\mathrm{dx2}\mathrm{dx2}+\mathrm{dx3}\mathrm{dx3}+\mathrm{dx4}\mathrm{dx4}$

$\mathrm{QuadraticFormSignature}\left(\mathrm{g2}\right)$

$\left[\left[\mathrm{D\_x2}\,\mathrm{D\_x3}\,\mathrm{D\_x4}\right]\,\left[\mathrm{D\_x1}\right]\,\left[\right]\right]$

$\mathrm{g3}≔\mathrm{evalDG}\left(2\mathrm{dx1}\&s\mathrm{dx2}+2\mathrm{dx3}\&s\mathrm{dx4}\right)$

$\mathrm{g3}:=\mathrm{dx1}\mathrm{dx2}+\mathrm{dx2}\mathrm{dx1}+\mathrm{dx3}\mathrm{dx4}+\mathrm{dx4}\mathrm{dx3}$

$\mathrm{QuadraticFormSignature}\left(\mathrm{g3}\right)$

$\left[\left[\mathrm{D\_x1}+\mathrm{D\_x2}\,\mathrm{D\_x3}+\mathrm{D\_x4}\right]\,\left[\mathrm{D\_x1}-\mathrm{D\_x2}\,\mathrm{D\_x3}-\mathrm{D\_x4}\right]\,\left[\right]\right]$

$\mathrm{g4}≔\mathrm{evalDG}\left(\mathrm{dx1}\&t\mathrm{dx1}+\mathrm{dx2}\&t\mathrm{dx2}-\mathrm{dx3}\&t\mathrm{dx3}\right)$

$\mathrm{g4}:=\mathrm{dx1}\mathrm{dx1}+\mathrm{dx2}\mathrm{dx2}-\left(\mathrm{dx3}\mathrm{dx3}\right)$

$\mathrm{QuadraticFormSignature}\left(\mathrm{g4}\right)$

$\left[\left[\mathrm{D\_x1}\,\mathrm{D\_x2}\right]\,\left[\mathrm{D\_x3}\right]\,\left[\mathrm{D\_x4}\right]\right]$

$\mathrm{QuadraticFormSignature}\left(\mathrm{g4}\,\mathrm{output}=''dimensions''\right)$

$\left[2\,1\,1\right]$

$\mathrm{QuadraticFormSignature}\left(\mathrm{g2}\,\left[\mathrm{D\_x1}\,\mathrm{D\_x2}\,\mathrm{D\_x3}\right]\right)$

$\left[\left[\mathrm{D\_x2}\,\mathrm{D\_x3}\right]\,\left[\mathrm{D\_x1}\right]\,\left[\right]\right]$

$\mathrm{QuadraticFormSignature}\left(\mathrm{g4}\,\left[\mathrm{D\_x1}\,\mathrm{D\_x3}\,\mathrm{D\_x4}\right]\right)$

$\left[\left[\mathrm{D\_x1}\right]\,\left[\mathrm{D\_x3}\right]\,\left[\mathrm{D\_x4}\right]\right]$

$\mathrm{DGsetup}\left(\left[\mathrm{x1}\,\mathrm{x2}\,\mathrm{x3}\,\mathrm{x4}\right]\,M\right)$

$\mathrm{frame\; name:\; M}$

$\mathrm{g5}≔\mathrm{evalDG}\left(a\mathrm{dx1}\&tensor\mathrm{dx1}-\mathrm{dx2}\&t\mathrm{dx2}+\mathrm{dx3}\&t\mathrm{dx3}+\mathrm{dx4}\&t\mathrm{dx4}\right)$

$\mathrm{g5}:=\left(a\mathrm{dx1}\right)\mathrm{dx1}-\left(\mathrm{dx2}\mathrm{dx2}\right)+\mathrm{dx3}\mathrm{dx3}+\mathrm{dx4}\mathrm{dx4}$

$\mathrm{QuadraticFormSignature}\left(\mathrm{g5}\right)$

$\mathrm{FAIL}$

$\mathrm{QuadraticFormSignature}\left(\mathrm{g5}\right)\mathbf{assuming}0<a$

$\left[\left[\mathrm{D\_x1}\&comma;\mathrm{D\_x3}\&comma;\mathrm{D\_x4}\right]\&comma;\left[\mathrm{D\_x2}\right]\&comma;\left[\right]\right]$

$\mathrm{QuadraticFormSignature}\left(\mathrm{g5}\right)\mathbf{assuming}a<0$

$\left[\left[\mathrm{D\_x3}\&comma;\mathrm{D\_x4}\right]\&comma;\left[\mathrm{D\_x1}\&comma;\mathrm{D\_x2}\right]\&comma;\left[\right]\right]$

$\mathrm{g6}≔\mathrm{evalDG}\left(a\mathrm{dx1}\&tensor\mathrm{dx1}+2\mathrm{dx1}\&s\mathrm{dx2}-\mathrm{dx2}\&t\mathrm{dx2}+\mathrm{dx3}\&t\mathrm{dx3}+\mathrm{dx4}\&t\mathrm{dx4}\right)$

$\mathrm{g6}:=\left(a\mathrm{dx1}\right)\mathrm{dx1}+\mathrm{dx1}\mathrm{dx2}+\mathrm{dx2}\mathrm{dx1}-\left(\mathrm{dx2}\mathrm{dx2}\right)+\mathrm{dx3}\mathrm{dx3}+\mathrm{dx4}\mathrm{dx4}$

$\mathrm{QuadraticFormSignature}\left(\mathrm{g6}\right)$

$\mathrm{FAIL}$

$\mathrm{infolevel}\left[\mathrm{QuadraticFormSignature}\right]≔2$

${\mathrm{infolevel}}_{\mathrm{DifferentialGeometry:-Tensor:-QuadraticFormSignature}}:=2$

$\mathrm{QuadraticFormSignature}\left(\mathrm{g6}\right)$

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

$\mathrm{FAIL}$

$\mathrm{infolevel}\left[\mathrm{QuadraticFormSignature}\right]≔0$

${\mathrm{infolevel}}_{\mathrm{DifferentialGeometry:-Tensor:-QuadraticFormSignature}}:=0$

$\mathrm{QuadraticFormSignature}\left(\mathrm{g6}\right)\mathbf{assuming}a<-1$

$\left[\left[\mathrm{D\_x3}\&comma;\mathrm{D\_x4}\right]\&comma;\left[\mathrm{D\_x1}\&comma;\mathrm{D\_x1}-\left(a\mathrm{D\_x2}\right)\right]\&comma;\left[\right]\right]$

$\mathrm{QuadraticFormSignature}\left(\mathrm{g6}\right)\mathbf{assuming}-1<a$

$\left[\left[\mathrm{D\_x1}+\mathrm{D\_x2}\&comma;\mathrm{D\_x3}\&comma;\mathrm{D\_x4}\right]\&comma;\left[\mathrm{D\_x2}\right]\&comma;\left[\right]\right]$