Define the matrix exponential \(e ^ { A }\) of a matrix A as follows: \(e ^ { A } \ = \ \sum _ { n = 0 } ^ { \infty } { \frac { A ^ { n } } { n ! } }\) Consider the following matrices
\[ M _ { x } = { \left[ \begin{array} { l l l } { 0 } & { 0 } & { 0 } \\ { 0 } & { 0 } & { - 1 } \\ { 0 } & { 1 } & { 0 } \end{array} \right] } , M _ { y } = { \left[ \begin{array} { l l l } { 0 } & { 0 } & { 1 } \\ { 0 } & { 0 } & { 0 } \\ { - 1 } & { 0 } & { 0 } \end{array} \right] } , M _ { z } = { \left[ \begin{array} { l l l } { 0 } & { - 1 } & { 0 } \\ { 1 } & { 0 } & { 0 } \\ { 0 } & { 0 } & { 0 } \end{array} \right] } .\tag{7} \]
Write \(\mathrm { T r } M _ { i } , M _ { i } ^ { T } , i = x , y , z\)
Calculate \(e ^ { \theta M _ { x } } , e ^ { \theta M _ { y } } , e ^ { \theta M _ { z } }\) , where \(\theta \in [ 0 , 2 \pi )\)
Give a geometrical interpretation of the transformations \(e ^ { \theta M _ { i } } , i = x , y , z\) acting on the vectors in \(\mathbb { R } ^ { 3 }\)