1.1.23 Geometric Visualization of the Inner Product

To vi­su­al­ize what an inner prod­uct of two $3 \times 3$ ma­tri­ces $\in U(3)$ might look like, we use the ex­am­ples for $G\left( X_{1} \right),G\left( X_{2} \right)$ ro­tat­ing the iden­ti­ty matrix $I_{3}$, with en­tries $(A,B,C)$ along its di­ag­o­nal, rep­re­sent­ing the ver­tices of the equi­lat­er­al tri­an­gle. Set­ting the ro­ta­tion angles $\angle X_{1} = \left( \frac{i\pi}{3} \right),\ \angle X_{2} = \left( - \frac{i\pi}{4} \right)$, this yields the fol­low­ing

and

Taking the inner prod­uct $\left\langle {\widehat{X}}_{1},{\widehat{X}}_{2} \right\rangle$

we find com­plex-valued $A,B$ vertex com­po­nents and a real-valued com­po­nent for the $C$ vertex. This is con­sis­tent with the effect of ro­tat­ing in the $X_{1}/X_{2}$-plane, for the $A,B$ ver­tices, having been on the com­plex axes of ro­ta­tion for $X_{1},X_{2}$ re­spec­tive­ly, while the $X_{3}$-plane did not act as a com­plex axis of ro­ta­tion on the $C$ vertex. In Figure 1.1.15, we have plot­ted both ro­ta­tions of the equi­lat­er­al tri­an­gle iden­ti­ty matrix in the $X_{1}/X_{2}$-plane.

The equi­lat­er­al tri­an­gle ${\widehat{X}}_{2} = \left( A_{2},B_{2},C_{2} \right)$ looks larger than ${\widehat{X}}_{1} = \left( A_{1},B_{1},C_{1} \right)$, but from the stand­point of the Hilbert-Schmidt norms, they are the same. This can be seen by first con­sid­er­ing $\left\| {\widehat{X}}_{1} \right\|$

then for $\left\| {\widehat{X}}_{2} \right\|$,

re­call­ing that $A,B,C \in \mathbb{R}^{+}$, and that $A = B = C$ rep­re­sent scalar mag­ni­tudes. Closer in­spec­tion also re­veals that the inner prod­uct $\left\langle {\widehat{X}}_{1},{\widehat{X}}_{2} \right\rangle = tr\left( {\widehat{X}}_{1}^{*}{\widehat{X}}_{2} \right)$, is es­sen­tial­ly the sum of the dot prod­ucts of the matched column vec­tors from ${\widehat{X}}_{1}$ and ${\widehat{X}}_{2}$, as

In this sec­tion, we have in­tro­duced the $U(3)$ sym­me­try group as the in­ter­con­nec­tion of three com­plex unit cir­cles which form a com­plex unit sphere. Ro­ta­tions in $U(3)$ are the result of group gen­er­a­tors acting on the equi­lat­er­al tri­an­gle as uni­tary linear trans­for­ma­tions. Hilbert spaces and their inner-prod­uct map­pings in­tro­duce com­plete vector spaces, a de­par­ture from the con­fines of Eu­clid­ean space.