Check that cycle similarity respects cycle product¶
Exercise I.6.24.i [1] $ \newcommand{\scalar}[3][]{\left\langle #2,#3 \right\rangle_{#1}} \newcommand{\Space}[3][]{\mathbb{#2}^{#3}_{#1}} \newcommand{\Space}[3][]{\mathbb{#2}^{#3}_{#1}} \newcommand{\cycle}[3][]{{#1 C^{#2}_{#3}}} \newcommand{\realline}[3][]{#1 R^{#2}_{#3}} \newcommand{\bs}{\breve{\sigma}} \newcommand{\zcycle}[3][]{#1 Z^{#2}_{#3}} \newcommand{\SL}[1][2]{\mathrm{SL}_{#1}(\Space{R}{})} \newcommand{\rs}{\sigma_r} \newcommand{\lvec}[1]{\overrightarrow{#1}} $ Show that the $\breve{\sigma}$-similarity $\breve{\sigma}$-preserves the orthogonality relation of cycles. More specifically, if $\cycle[\dot]{s}{\breve{\sigma}}$ and $\cycle[\ddot]{s}{\breve{\sigma}}$ are matrix similarity of cycles $\cycle{s}{\breve{\sigma}}$ and $\cycle[\hat]{s}{\breve{\sigma}}$, respectively, with the cycle $\cycle[\tilde]{s_1}{\breve{\sigma}}$, then \begin{equation} \scalar{\cycle[\dot]{s}{\breve{\sigma}}}{\cycle[\ddot]{s}{\breve{\sigma}}}= \scalar{\cycle{s}{\breve{\sigma}}}{\cycle[\hat]{s}{\breve{\sigma}}} (\det \cycle[\tilde]{s}{\breve{\sigma}})^2. \end{equation} Hint. Note that $\cycle[\tilde]{s}{\breve{\sigma}}\overline{\cycle[\tilde]{s}{\breve{\sigma}}}= -\det(\cycle[\tilde]{s}{\breve{\sigma}}) I$, where $I$ is the identity matrix. This is a particular case of the \emph{Vahlen condition} see [FillmoreSpringer90a, Prop.~2]. Thus, we have \begin{equation} \cycle[\dot]{s}{\breve{\sigma}}\overline{\cycle[\ddot]{s}{\breve{\sigma}}}= -\cycle[\tilde]{s_1}{\breve{\sigma}}{\cycle{s}{\breve{\sigma}}}\overline{\cycle[\hat]{s}{\breve{\sigma}}} \overline{\cycle[\tilde]{s_1}{\breve{\sigma}}} \cdot \det \cycle[\tilde]{s}{\breve{\sigma}}. \end{equation} The final step uses the invariance of the trace under the matrix similarity. A CAS calculation is also provided.
Solution. This is not only a solution for the exercise but also a test for the library. We define another set of cycle parameters
from init_cycle import *
k2=realsymbol("k2")
l2=realsymbol("l2")
n2=realsymbol("n2")
m2=realsymbol("m2")
C2=cycle2D(k2,[l2,n2],m2,e)
Complex matrices corresponding to the cycles
Mat=matrix([[l+I*sign2*n,-m],[k,-l+I*sign2*n]])
Mat1=matrix([[l1+I*sign1*n1,-m1],[k1,-l1+I*sign1*n1]])
Mat2=matrix([[l2+I*sign1*n2,-m2],[k2,-l2+I*sign1*n2]])
First we calculate similarity of respective matrices
Res7=(-Mat*Mat1.conjugate()*Mat).evalm().expand()
Res8=(-Mat*Mat2.conjugate()*Mat).evalm().expand()
Check that the matrix similarity preserves the inner product
"Cycle similarity respects cycle product (directly): %s" % \
((Res7*Res8.conjugate()).evalm().trace()-\
(Mat1*Mat2.conjugate()).evalm().trace()*pow(Mat.determinant(),2)).normal().is_zero()
Then we calculate cycle similarity from the libary
C3=C1.cycle_similarity(C,e,diag_matrix([1,sign1]),diag_matrix([1,sign2]))
C4=C2.cycle_similarity(C,e,diag_matrix([1,sign1]),diag_matrix([1,sign2]))
Check that the matrix similarity preserves the inner product (library)
"Cycle similarity respects cycle product (library): %s" % \
(C3.cycle_product(C4,e,diag_matrix([1,sign1]))\
-C1.cycle_product(C2,e,diag_matrix([1,sign1]))*pow(C.hdet(),2))\
.subs({pow(sign1,2) : 1},subs_options.algebraic)\
.subs({pow(sign2,2) : 1},subs_options.algebraic).normal().is_zero()
This notebook is a part of the MoebInv notebooks project [2] .
References¶
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.
Vladimir V. Kisil, MoebInv notebooks, 2019.