1.4.11 Geometric Visualization of Matrix Group Conics

In this final sec­tion, we ex­plore the geom­e­try of matrix group conics through vi­su­al­iza­tion. The reader is most likely fa­mil­iar with the Carte­sian rep­re­sen­ta­tion of the unit circle $x^{2} + y^{2} = 1$, and this re­lates the in­vari­ant unit dis­tance from the origin $(0,0)$ to the co­or­di­nates $(x,y)$, which to­geth­er, form the equa­tion of a circle. In the 2-di­men­sion­al matrix form, the de­ter­mi­nant of the unit circle's polar rep­re­sen­ta­tion gives us the trigono­met­ric de­ter­mi­nant form

and com­par­ing this to the Carte­sian form, we see that $x^{2} = \cos^{2}(\theta)$ and $y^{2} = \sin^{2}(\theta)$. In more gen­er­al terms, we can state the Carte­sian form and how it re­lates to its radius $r$, as

and in the polar form

A sim­i­lar Carte­sian-to-polar form exists for hy­per­bol­ic geom­e­try, where we start by taking the de­ter­mi­nant of the $2 \times 2$ matrix rep­re­sent­ing the unit hy­per­bo­la

which in Carte­sian co­or­di­nates trans­lates, as $x^{2} = \cosh^{2}(\beta),\ y^{2} = \sinh^{2}(\beta)$, having the de­ter­mi­nant

The unit hy­per­bol­ic rep­re­sen­ta­tions also present with in­vari­ant unit length for their matrix de­ter­mi­nants, which is typ­i­cal­ly de­not­ed as a radial dis­tance from the origin $\rho$ in the hy­per­bol­ic trigono­met­ric rep­re­sen­ta­tion

which, in Carte­sian form, is ex­pressed as

The cone is clas­si­cal­ly rep­re­sent­ed by lines in real Eu­clid­ean space, while the rep­re­sen­ta­tion for our matrix groups will be more de­tailed. We recall the two-di­men­sion­al matrix group rep­re­sen­ta­tion for the cone number $( - 1)^{\left( \frac{1}{4} \right)}$

which has the de­ter­mi­nant

Here, we also see the in­vari­ant unit length, by way of the unit-valued de­ter­mi­nants$.$It may not be ob­vi­ous in the form above, but this cone matrix group is a bal­anced rep­re­sen­ta­tion of the unit circle and the unit hy­per­bo­la. Con­sid­er­ing the com­plex co­or­di­nates $( - 1)^{\frac{1}{4}} = \left( \frac{1}{\sqrt{2}} + \frac{i}{\sqrt{2}} \right),$our cone group ex­am­ple rep­re­sents the ex­po­nen­tial func­tion $e^{\alpha\left( \frac{1}{\sqrt{2}} + \frac{i}{\sqrt{2}} \right)}$, where it be­comes ap­par­ent that it is equal parts real ex­po­nent and imag­i­nary ex­po­nent, ev­i­denced by rewrit­ing the ex­po­nen­tial, as

Now, we can rep­re­sent this as the prod­uct of two matrix groups

and when con­sid­er­ing the de­ter­mi­nants of these matrix groups, we find

which is equiv­a­lent to

In the de­ter­mi­nant prod­ucts above, the angle $\alpha$ is down­scaled by the factor $\frac{1}{\sqrt{2}}$ when com­pared to the in­di­vid­ual unit circle and hy­per­bo­la, making it­er­a­tive cir­cu­lar ro­ta­tions for each $\alpha = \left( \sqrt{2} \right)2\pi n$, $n\in \mathbb{Z}.$We are tempt­ed to con­sid­er the re­la­tion­ship above in terms of their Carte­sian forms, rep­re­sen­ta­tive of the unit hy­per­bo­la and unit circle

This Carte­sian form does work to give us the first glimpse at what the prod­uct of a hy­per­bo­la and circle might look like when com­pared to the stand­alone forms of the unit hy­per­bo­la and circle (Figure 1.4.21).

In the graph­i­cal rep­re­sen­ta­tion of the equa­tion ${(x}^{4} - y^{4}) = 1$, the unit cone does take on the ap­pear­ance of the as­ymp­totes of the unit hy­per­bo­la, seen as it ex­tends fur­ther from the bound­ary of the unit disc. Shown in the image, is the unit hy­per­bo­lae (red), the unit cone's curves/as­ymp­totes (black), the unit circle (blue), and the hy­per­bol­ic as­ymp­totes (green).

To ensure the Carte­sian cone num­bers are math­e­mat­i­cal­ly con­sis­tent, we begin with the orig­i­nal prod­uct of the $2 \times 2$ ma­tri­ces

to ex­press these by the fol­low­ing prod­uct of matrix ex­po­nen­tials

Taking the de­ter­mi­nant of the re­sul­tant matrix above,

to then con­sid­er this result in the Carte­sian form of the unit hy­per­bo­la raised to the power $\left( \frac{2}{\sqrt{2}} + \frac{2i}{\sqrt{2}} \right)$

To gain an in­tu­ition for what this func­tion looks like, we plot the imag­i­nary and real por­tions in three-di­men­sions in Figure 1.4.22, and as con­tour plots in Figure 1.4.23.

We can see a cone struc­ture that emerges in the real and imag­i­nary-valued three-di­men­sion­al/con­tour plots. Turn­ing to the cone el­e­ments of the sym­me­try group, we start by taking a tan­gent space el­e­ment $\Delta X$, like that we worked with for the $\mathfrak{su}(2)$ sub­group, and define $\Delta X$, as

The next step is to con­sid­er how to arrive at the entire tan­gent space for the $X_{1}$-ori­ent­ed double cone. To do this, we will have to rotate the matrix $\Delta X$ in the $X_{2},X_{3}$-plane, by using $X_{1}$ as the axis, which is ex­act­ly what the $X_{1}$-el­e­ment $\in U(3)$ does! De­not­ing the ex­po­nen­ti­at­ed $X_{1}$-type el­e­ment $\in U(3)$, as $E,$and using the angle ar­gu­ment $\theta_{2}$, we have

The prod­uct,$E \times (\theta_{1}\Delta X)$, for $\theta_{1}$, the angle ar­gu­ment of $\Delta X$, yields the group form that en­com­pass­es the whole tan­gent space path of the $X_{1}$ cone sub­group

To make it clear­er as to what the matrix prod­uct $R^{k} = \ E^{k} \times \left( \theta_{1}\Delta X \right)$ is, we set $\theta_{2} = \frac{\pi}{4}$, then per­form eight it­er­a­tive ro­ta­tions of $\theta_{1}\Delta X$, making a full ro­ta­tion of the disc bound­ary formed as a per­pen­dic­u­lar slice of the cone

From the ro­ta­tion­al it­er­a­tions above, the ev­i­dence that a full rev­o­lu­tion around the cone tan­gent space has oc­curred, comes by way of the return to the start­ing el­e­ment at ${R^{8} = E}^{8} \times \left( \theta_{1}\Delta X \right) = \Delta X$. One way to con­sid­er the graph­i­cal rep­re­sen­ta­tion of the ro­tat­ing $\Delta X$-el­e­ment, is by the fol­low­ing series of fig­ures, where we start with the $\Delta X$-el­e­ment and pro­ceed up to the $R^{4}$ ro­ta­tion.

By the ro­ta­tion rep­re­sent­ed in Figure 1.4.26b, we have ar­rived at the neg­a­tive form of our ini­tial $\Delta X$-el­e­ment, which is dis­cerned by the vector's ar­row­head as com­pared to the $\Delta X$-el­e­ment (Figure 1.4.24). The fig­ures rep­re­sent the ro­ta­tion­al axis $X_{1}$, as fixed, but when look­ing at the it­er­a­tions $R^{k}$, the axis of ro­ta­tion also moves along the unit circle. The result of this dy­nam­ic axis is that we have an­tipo­dal vector rep­re­sen­ta­tions of all el­e­ments in the tan­gent space. The el­e­ments gen­er­at­ed from the $R^{k}$-ro­ta­tions are very sim­i­lar to those el­e­ments which gen­er­ate the $\mathfrak{su}(2)$ Lie al­ge­bra. We pursue this sim­i­lar­i­ty, by eval­u­at­ing prod­ucts of the $R^{k}$-el­e­ments, where we omit the angle vari­able $\theta_{1}$, to rep­re­sent sim­i­lar forms to those of the $\mathfrak{su}(2)$ el­e­ments

In the fol­low­ing prod­ucts of the $\mathcal{u}_{k}$-el­e­ments, the bottom-right $2 \times 2$ sub­ma­tri­ces hold to the $\mathfrak{su}(2)$ form, while the axis entry ro­tates in the axial-plane

The name as­sign­ment for these matrix prod­ucts $\mathcal{u}_{k}$, is based on the bottom-right $2 \times 2$ sub­ma­tri­ces. The axes en­tries $a_{1,1}$ remain un­changed if con­sid­er­ing the com­mu­ta­tion op­er­a­tion of two el­e­ments $\left\lbrack \mathcal{u}_{j},\mathcal{u}_{k} \right\rbrack$. We can also see the an­tipo­dal re­la­tion­ships by con­sid­er­ing that the vector el­e­ments $R^{2},R^{8}$ are re­spec­tive­ly an­tipo­dal to the el­e­ments $R^{4},R^{6}$. The matrix prod­ucts of an­tipo­dal el­e­ments can be seen to gen­er­ate the $3 \times 3$ $\mathfrak{su}(2)$ Lie al­ge­bras $\mathcal{u}_{1},\mathcal{u}_{2}$

To gen­er­al­ize the con­cept of an­tipo­dal el­e­ments, we rep­re­sent the phase shift $\pm \pi$ ra­di­ans, from the pre­vi­ous ro­ta­tion matrix $R$, by using the map­ping, $\mathcal{AP}(R) \rightarrow \ - R$

Re­turn­ing to the $R$-ro­ta­tion matrix, we find its de­ter­mi­nant equals

for which, we see that any el­e­ment $\in X_{1}$-cone tan­gent space has a de­ter­mi­nant of $- \theta_{1}^{3}e^{i\theta_{2}}$, since $\cos^{2}{\left( \theta_{2} \right) + \sin^{2}\left( \theta_{2} \right)} = 1.$Set­ting $\theta_{1} = 1$, the de­ter­mi­nant sim­pli­fies to

where the volume of the $X_{1}$-tan­gent space ring is simply $- e^{i\theta_{2}}$, and for the case of our $R^{k}$-ro­ta­tions, for $\theta_{2} = \frac{\pi}{4}$, we find that $- e^{\left( i\frac{\pi}{4} \right)} = - ( - 1)^{\left( \frac{1}{4} \right)}$.

Next, would like to con­sid­er the di­ver­gence and curl of the $R$-matrix. To do this we can de­com­pose $dR$ into its sym­met­ric, $SR$, and an­ti­sym­met­ric, $AR$, parts

where

The di­ag­o­nal matrix $SR$ has a trace rep­re­sen­ta­tive of the di­ver­gence of the $R$-matrix group, while $AR$ rep­re­sents the curl-like an­ti­sym­met­ric por­tion of the $R$-matrix group

The sym­met­ric matrix $AS^{'}$ is now trace-free

and we can ex­press $dR$, as

Con­cern­ing the curl matrix $\nabla \times R$, a ro­ta­tion is present, match­ing the cir­cu­lar ring that traces out the $X_{1}$-cone tan­gent space, and is as ex­pect­ed. For the di­ver­gence matrix $\nabla \bullet R$, set­ting, $\theta_{1} = 1$, and then plot­ting the cosine func­tions, re­veals real values $\frac{\pi}{4}$ ra­di­ans out of phase, and tandem in-phase su­per­im­posed imag­i­nary values/waves, seen in the figure below.

$(\cos(\theta)( - 1)^{\frac{1}{4}}$ (blue), $\cos(\theta)( - 1)^{\frac{3}{4}}$ (red), and imag­i­nary-valued bottom.

In­tu­itive­ly, the mean­ing of the di­ver­gence matrix $\nabla \bullet R$

is that the di­ver­gence os­cil­lates back and forth by ${i\theta}_{1}e^{i\theta_{2}}$ in the $X_{1}$-plane, by $( - 1)^{\left( \frac{1}{4} \right)}\theta_{1}\cos\left( \theta_{2} \right)$ in the $X_{2}$-plane, and by $( - 1)^{\left( \frac{3}{4} \right)}\theta_{1}\cos\left( \theta_{2} \right)$ in the $X_{3}$-plane. Set­ting $\theta_{1} = 1,\theta_{2} = \frac{\pi}{4}$, as we did in the de­ter­mi­nant ex­am­ple, we find that the $X_{1}$- plane di­verges by $ie^{\left( \frac{\pi}{4} \right)} = ( - 1)^{\left( \frac{3}{4} \right)}$, the $X_{2}$-plane by $( - 1)^{\left( \frac{1}{4} \right)}\cos\left( \frac{\pi}{4} \right) = \left( \frac{1}{2} + \frac{i}{2} \right)$, and the $X_{3}$-plane by $( - 1)^{\left( \frac{3}{4} \right)}\cos\left( \frac{\pi}{4} \right) = \left( - \frac{1}{2} + \frac{i}{2} \right).$Taking the mod­u­lus of each of these com­plex values yields, $\left| X_{1} \right| = 1,\left| X_{2} \right| = \frac{1}{\sqrt{2}},$and $\left| X_{3} \right| = \frac{1}{\sqrt{2}}$, which make in­tu­itive sense since the moduli of the $X_{2}X_{3}$-plane values are a dis­tance of $\frac{1}{\sqrt{2}}$ away from the $X_{1}$-axis. If we were to re­de­fine our $X_{2}X_{3}$-plane ro­ta­tion matrix, $E$, to

the ro­ta­tion axis would now be real-valued, ad­mit­ting all values $\theta_{1}$ $\in \mathbb{R}$, and the $R$-matrix de­ter­mi­nant be­comes

We con­sid­er the Carte­sian equa­tion of the in­fi­nite cone

whose para­met­ric form can be found by solv­ing for $z$

then let­ting

So, by set­ting $A = B = \theta_{1}^{3}$, and defin­ing, $X = \theta_{1}^{\left( \frac{3}{2} \right)}e^{\left( \frac{\theta_{2}}{2} \right)}\cos\left( \theta_{2} \right),\ Y = \theta_{1}^{\left( \frac{3}{2} \right)}{e^{\left( \frac{\theta_{2}}{2} \right)}\sin}\left( \theta_{2} \right),\ Z = \theta_{1}^{\left( \frac{3}{2} \right)}e^{\left( \frac{\theta_{2}}{2} \right)}$, the re­sult­ing equa­tion rep­re­sents the in­fi­nite Carte­sian unit cone, plot­ted in Figure 1.4.28

Con­cern­ing the in­fi­nite cone matrix group, an im­por­tant dis­tinc­tion to make is for non-zero $\theta_{1}$ values, the cone will not come to a zero point at the center, and once the $R$-matrix is ex­po­nen­ti­at­ed into the group-like form, there will be a non-zero geo­met­ric sphere at the center, which re­mains in the tan­gent space. An ex­am­ple would be to set $\theta_{1} = 1$, which yields the fol­low­ing de­ter­mi­nant for $R$

and the define $X = e^{\left( \frac{\theta_{2}}{2} \right)}\cos\left( \theta_{2} \right),\ Y = e^{\left( \frac{\theta_{2}}{2} \right)}\sin\left( \theta_{2} \right),\ Z = e^{\left( \frac{\theta_{2}}{2} \right)}$, the result is

For all values $\theta_{2} = \lbrack 0,\infty)$, the in­fi­nite cone equa­tion above gives the result $Z^{2}$, which is $\geq 1$, and for values less than zero, $Z^{2}$ ap­proach­es zero only as $\theta_{2} \rightarrow - \infty.$Al­though the form $X^{2} + Y^{2} = Z^{2}$ is al­ge­braical­ly cor­rect, the same $\theta_{2}$ vari­able is present in all the com­po­nents, and its path will travel equal in mag­ni­tude por­tions of the real and imag­i­nary forms on the cone. The prod­uct of the $E$-matrix and $\exp\left( \theta_{1}\Delta X \right)$, yields a matrix group prod­uct $R \times e^{\theta_{1}\Delta X}$, which en­com­pass­es both $\theta_{1},\theta_{2}$ vari­ables

By taking the de­ter­mi­nant of $E \times e^{\theta_{1}\Delta X}$, it can be seen clear­ly that this is the prod­uct of the unit hy­per­bo­la and the unit circle

Since $\left( {\cos h^{2}}{\left( \theta_{1} \right) - {\sin h^{2}}\left( \theta_{1} \right)} \right),\left( \cos^{2}{\left( \theta_{2} \right) + \sin^{2}\left( \theta_{2} \right)} \right) = 1$, we can ex­press the result of the de­ter­mi­nant, as

The $e^{\theta_{1} + {i\theta}_{2}}$ func­tion can be con­sid­ered in terms of its real-valued com­po­nent, rep­re­sent­ing the mag­ni­tude of the func­tion along the real $X_{1}$-axis and the imag­i­nary-valued com­po­nent as the cir­cu­lar ro­ta­tion in the $X_{2}X_{3}$-plane. We can con­sid­er this de­ter­mi­nant value as a non-axis number for $\theta_{1},\theta_{2} \neq 0$, which is con­sis­tent with its cone number de­f­i­n­i­tion. To vi­su­al­ly con­sid­er the de­ter­mi­nant, we plot in Figure 1.4.29 the func­tion $e^{\theta_{1} + {i\theta}_{2}}$, where the $X_{1}$-plane is coming out of the page, and the $X_{2}X_{3}$- plane is the two-di­men­sion­al rep­re­sen­ta­tion of the flat page. For larger values of the real com­po­nent $\theta_{1}$, we would see an in­creased dis­tance be­tween the arms of the spi­rals on the cone and, for small­er values of $\theta_{1}$, tight­ly packed spi­rals on the cone would occur.

So far, we have been con­sid­er­ing the cone sec­tion for the $X_{1}$-plane. The rep­re­sen­ta­tions are sim­i­lar for the cone sec­tions in the $X_{2}$and $X_{3}$-planes, which trace out cir­cu­lar tan­gent paths in the re­spec­tive $X_{1}X_{3}$,$X_{1}X_{2}$-planes. Sim­i­lar rep­re­sen­ta­tions are gen­er­at­ed for the $X_{2}$-plane cone, as

and the $X_{3}$-plane cone, as

The unit sphere at the center of the conic is easily vi­su­al­ized as the in­ter­sec­tion of three cir­cu­lar rings, rep­re­sent­ing the com­plex unit circle in its three or­thog­o­nal planes $X_{1},X_{2},X_{3}$. This sphere is the $U(3)$ group rep­re­sen­ta­tion. In terms of the matrix group conic's hy­per­bo­lae, we can en­vi­sion them as the result of each unit circle being opened and re­flect­ed from the imag­i­nary axes to the real axes in the $X_{1},X_{2},X_{3}$-planes, by way of a change in ec­cen­tric­i­ty. The hy­per­bol­ic analog to the $U(3)$-sphere is com­posed of unit hy­per­bo­lae, as three con­vex­ly ori­ent­ed hy­per­bol­ic curves in each plane, facing away from the origin in the pos­i­tive and neg­a­tive di­rec­tions. In Figure 1.4.30, we graph the geo­met­ric arrange­ment of the unit cir­cles and unit hy­per­bo­lae with re­spect to one an­oth­er.

and the unit hy­per­bo­lae in­ter­sect­ing the sphere at $e^{0}$ in the $X_{1},X_{2},X_{3}$- planes.

Math­e­mat­i­cal­ly, the hy­per­bo­la rep­re­sents an open and ex­tend­ed geo­met­ric curve, while the $U(3)$ sphere is closed and non-ex­tend­ed. The cone can then be thought of as the per­fect bal­ance be­tween the open and closed geome­tries of the circle and hy­per­bo­la in $\mathbb{C}^{2}$. We can en­vi­sion it as the per­fect bal­ance be­tween the $U(3)$ sphere and the hy­per­ge­om­e­try of the six-di­men­sion­al unit hy­per­bo­lae.

unit hy­per­bo­la and the six-di­men­sion­al com­plex unit sphere.

In Figure 1.4.31, we plot the $X_{1}$-in­fi­nite unit cone, ori­ent­ed in re­la­tion­ship to the hy­per­bol­ic and spher­i­cal geome­tries. The cones rep­re­sent the as­ymp­tot­ic bound­aries of the hy­per­bo­lae, which only ap­proach this bound­ary at in­fin­i­ty. For the geome­tries ex­tend­ing beyond the unit in­vari­ant lengths, we plot the cone and the hy­per­bo­la in the $X_{1}$- plane, Figure 1.4.32, where blue rep­re­sents the $X_{1}$-cone group, grey, the $X_{1}$-hy­per­bo­la, and dark grey, the tan­gent sphere cen­tered at the origin.

The matrix group conic sec­tions become ex­tend­ed through the ex­po­nen­tial map­ping of the tan­gent spaces on the sur­face of the sphere. The tan­gent sphere is the matrix group rep­re­sen­ta­tion of a sin­gu­lar­i­ty sit­u­at­ed at the origin, which con­tains all the in­for­ma­tion of the matrix groups, con­veyed by the Lie al­ge­bra's in­for­ma­tion on its sur­face. Math­e­mat­i­cal­ly, values are bound to the var­i­ous geome­tries that arise with the group ex­po­nen­ti­a­tion of Lie al­ge­bras and these values can be bound to cer­tain sub­groups of the group. An ex­am­ple of this would be bound­ed­ness on the six-di­men­sion­al hy­per­bo­lae $\in \mathbb{R}^{6}$, which arises from the matrix ex­po­nen­ti­a­tion of the $D_{3}/S_{3}$ Lie al­ge­bra, where boosts can elicit move­ment in the ex­tend­ed sub­space of the real num­bers, but this occurs in a dis­tinct sub­space of the matrix group conic sec­tions.