None ex.I.5.25
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Types of the Cauchy-Schwartz inequality is inherited by the linear span in Krein spaces

Exercise I.5.25 Let two vectors $x$ and $y$ in an indefinite product space satisfy either of the inequalities: \begin{equation} \newcommand{\scalar}[2]{\langle #1, #2 \rangle} \scalar{x}{y}\scalar{y}{x} < \scalar{x}{x} \scalar{y}{y}\qquad\text{or}\qquad \scalar{x}{y}\scalar{y}{x} > \scalar{x}{x} \scalar{y}{y}. \end{equation} Then, any two non-collinear vectors $z$ and $w$ from the real linear span of $x$ and $y$ satisfy the corresponding inequality: \begin{equation} \scalar{z}{w}\scalar{w}{z} < \scalar{z}{z} \scalar{w}{w}\qquad\text{or}\qquad \scalar{z}{w}\scalar{w}{z} > \scalar{z}{z} \scalar{w}{w}. \end{equation} The equality $\scalar{x}{y}\scalar{y}{x} = \scalar{x}{x} \scalar{y}{y}$ always implies $\scalar{z}{w}\scalar{w}{z} = \scalar{z}{z} \scalar{w}{w}$.

Solution. Only GiNaC is required for this exercise.

In [1]:
from cycle import *

            Python wrappers for MoibInv Library
     ---------------------------------------------
Please cite this software as
V.V. Kisil, MoebInv: C++ libraries for manipulations in non-Euclidean geometry, SoftwareX, 11(2020),100385. doi:10.1016/j.softx.2019.100385.
     ---------------------------------------------

We define basic variables and elements of the linear spans.

In [2]:
A=realsymbol("A")
B=realsymbol("B")
t=realsymbol("t")
s=realsymbol("s")
E1=t*A+(1-t)*B # First linear span 
E2=s*A+(1-s)*B # Second linear span

The following symbols will be used in the substitutions:

We make E1*E2 with the replacement for indefinite products

In [3]:
A2=realsymbol("<A,A>", "\\langle A,A\\rangle") # represents indefinite product <A,A>
B2=realsymbol("<B,B>", "\\langle B,B\\rangle") # represents indefinite product <B,B>
AB=realsymbol("<A,B>", "\\langle A,B\\rangle") # represents indefinite product <A,B>
A2xB2=realsymbol("<A,A>*<B,B>", "\\langle A,A\\rangle \\langle B,B\\rangle") # represents product <A,A><B,B>
t=realsymbol("t")
s=realsymbol("s")

Second copy of E1*E1 with the replacement for indefinite products

In [4]:
Pr1=(E1*E2).expand().subs({A*A : A2, B*B : B2}).subs({A*B : AB},subs_options.algebraic)
Latex(f"${Pr1}$")
Out[4]:
$ s \langle B,B\rangle t-2 \langle A,B\rangle s t+ \langle A,B\rangle s- s \langle B,B\rangle+ s \langle A,A\rangle t+ \langle A,B\rangle t+\langle B,B\rangle- \langle B,B\rangle t$
In [5]:
Pr2=(E1*E1).expand().subs({A*A : A2, B*B : B2}).subs({A*B : AB},subs_options.algebraic)
Latex(f"${Pr2}$")
Out[5]:
$ \langle A,A\rangle t^{2}+2 \langle A,B\rangle t+ \langle B,B\rangle t^{2}+\langle B,B\rangle-2 \langle B,B\rangle t-2 \langle A,B\rangle t^{2}$

The third copy of E2*E2 with the replacement for indefinite products

In [6]:
Pr3=(E2*E2).expand().subs({A*A : A2, B*B : B2}).subs({A*B : AB},subs_options.algebraic)
Latex(f"${Pr3}$")
Out[6]:
$-2 \langle A,B\rangle s^{2}+2 \langle A,B\rangle s-2 s \langle B,B\rangle+ s^{2} \langle B,B\rangle+\langle B,B\rangle+ s^{2} \langle A,A\rangle$

We output the result of calculations:

In [7]:
Latex(f"for vectors $E_1={E1}$ and $E_2={E2}$ we have")
Out[7]:
for vectors $E_1= A t- B {(-1+t)}$ and $E_2=- {(-1+s)} B+ s A$ we have
In [8]:
Latex(r"$\langle E_1,E_2\rangle \langle E_2,E_1\rangle - \langle E_1,E_1\rangle \langle E_2,E_2\rangle =$"
f"${((Pr1*Pr1-Pr2*Pr3).expand().normal().subs({A2*B2 : A2xB2}, subs_options.algebraic)).collect(AB).collect(A2xB2)}$")
Out[8]:
$\langle E_1,E_2\rangle \langle E_2,E_1\rangle - \langle E_1,E_1\rangle \langle E_2,E_2\rangle =$$ \langle A,A\rangle \langle B,B\rangle {(2 s t-s^{2}-t^{2})}- \langle A,B\rangle^{2} {(2 s t-s^{2}-t^{2})}$

Thus, the sign of $ \newcommand{\scalar}[2]{\langle #1, #2 \rangle} \scalar{E_1}{E_2}\scalar{E_2}{E_1} - \scalar{E_1}{E_1} \scalar{E_2}{E_2} $ is exactly the same as $\scalar{A}{B}\scalar{B}{A} - \scalar{A}{A} \scalar{B}{B}$.

This notebook is a part of the MoebInv notebooks project [2] .

References

  1. Vladimir V. Kisil. Geometry of Möbius Transformations: Elliptic, Parabolic and Hyperbolic Actions of $SL_2(\mathbb{R})$. Imperial College Press, London, 2012. Includes a live DVD.

  2. Vladimir V. Kisil, MoebInv notebooks, 2019.

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