1.1.7 Group Exponentiation

The el­e­ments in the groups dis­cussed so far have been finite. We can ex­po­nen­ti­ate ma­tri­ces sim­i­lar­ly to how it is done for single vari­ables, making in­fi­nite de­grees of free­dom for ro­ta­tions in $\mathbb{R}^{n}\ and\ \mathbb{C}^{n}$. We define ex­po­nen­ti­a­tion for a matrix M by

where $\frac{M}{0!}$ equals the iden­ti­ty matrix with the same degree as M. We use the ex­am­ple

which would result in the fol­low­ing ex­po­nen­tial map­ping ex­pan­sion

Group­ing the above terms by ad­di­tion, we find the in­fi­nite series for trigono­met­ric sine and cosine func­tions appear

For small­er square ma­tri­ces $n \times n$, like the above, ex­po­nen­ti­a­tion is fea­si­ble, but for larger ma­tri­ces this be­comes chal­leng­ing, and many al­go­rithms exist to cal­cu­late them. For a di­ag­o­nal­iz­able matrix $M$, it is pos­si­ble to de­com­pose it into its eigen­val­ues arranged on the di­ag­o­nal matrix D, and by the re­spec­tive eigen­vec­tors of $M$, form­ing the or­tho­nor­mal column vec­tors in a matrix V. The matrix M can then be rep­re­sent­ed as

Using the matrix M from above, we can de­com­pose it into its eigen­vec­tors and eigen­val­ues by the de­fined ma­tri­ces$\ V,D$, to ex­press the de­com­po­si­tion of $M$, as

Using this de­com­po­si­tion, the matrix ex­po­nen­tial be­comes easier to cal­cu­late since ex­po­nen­ti­a­tion is then only re­quired for di­ag­o­nal matrix D. The matrix ex­po­nen­tial $e^{M}$ can now be ex­pressed in terms of the de­com­po­si­tion

At this point, we can apply what we have learned to­wards a matrix group ex­am­ple. We start with a simple rep­re­sen­ta­tion of the abelian cyclic group $Z_{3}$, where abelian de­fines that the matrix prod­ucts com­mute. The group M is de­fined as the set of three per­mu­ta­tions

for which, $m$ serves as our iden­ti­ty per­mu­ta­tion, all el­e­ments are com­mu­ta­tive (abelian), in­vers­es exist for each el­e­ment, and the group is closed under mul­ti­pli­ca­tion. Ex­am­ples of the group prop­er­ties for $M$ are

Iden­ti­ty:

In­verse:

Clo­sure:

Com­mu­ta­tive:

We are con­sid­er­ing a cyclic group, where the prod­ucts of the per­mu­ta­tions $m_{2}{,m}_{3}$ cycle through all the el­e­ments of the group. The per­mu­ta­tions $m_{2}{,m}_{3}$ will also cycle by the linear trans­for­ma­tion of the el­e­ments of a vector when left mul­ti­plied. Taking a vector $v = \lbrack y_{1},y_{2},y_{3}\rbrack'$ , we then left mul­ti­ply by $m_{2}$, find­ing

So, we see that for the prod­uct $m_{2}v$, the el­e­ments in $v$ have cycled upward by the linear trans­for­ma­tion$.$Fur­ther, for each $m_{j} \in M$, we can define some $y_{j}m_{j} = Y_{j}$, for $y_{j}\in \mathbb{R}$, to yield

Each $y_{j}$ value is an ar­gu­ment of the $Y_{j}$ ma­tri­ces above and each $Y_{j}$ can then be ex­po­nen­ti­at­ed to form a matrix group rep­re­sen­ta­tion. For the ex­am­ple $e^{Y_{2}}$, $Y_{2}$ is di­ag­o­nal­iz­able, there­fore, it can be rep­re­sent­ed as

for $V_{2}$, the eigen­vec­tor matrix of $Y_{2},$and $D_{2}$, the di­ag­o­nal matrix of eigen­val­ues of $Y_{2}$. The sim­pli­fied ex­po­nen­tial map­ping $e^{Y_{2}}$, then be­comes

Cal­cu­lat­ing $V_{2}$ and $D_{2}$, gives us

and using these de­com­po­si­tion com­po­nents, the matrix ex­po­nen­tial $e^{Y_{2}}$ will have the matrix group rep­re­sen­ta­tion,

In the matrix group above, we now have an in­fi­nite number of linear trans­for­ma­tions based on the abil­i­ty to set an in­fi­nite number of values for the ar­gu­ment $y\in \mathbb{R}$. The matrix group acts on $v,$as we showed pre­vi­ous­ly by left mul­ti­pli­ca­tion, and the trans­for­ma­tion yields a result that re­mains $\in \mathbb{R}^{3}$