Random Thought

We can define Note that we have and Triple product expansion We will show the above with the triple product expansion: Proof: (1)   Similarly for the and components. Proof of Note that for any     thus Proof of For any (2)   Thus,

The rotation matrix for rotating an object along normal direction with angle is given by where such that We can easily validate that the equation is correct, note that as desired. And for any vector perpendicular to as desired as well. Compute and from  Note that  Thus, . Moreover, since , we can compute as…

Lie algebra is a vector space $latex g$ with a map $latex [\cdot, \cdot]:g\times g \rightarrow g$ such that $latex [\cdot,\cdot]$ is bilinear $latex [x,x]=0 $ Jacobi inequality: $latex [x,[y,z]]+[y,[z,x]]+[z,[x,y]]=0,\forall x,y,z$. Note that 2) $latex \Rightarrow [x,y]=-[y,x]$ since $latex 0=[x+y,x+y]=[x,x]+[y,y]+[x,y]+[y,x]$. Note that the converse is true most of time as well since that implies $latex…

  The “S” in stands for special, meaning that , then . . The condition obviously requires . It turns out that the converse is true as well and so as shown in the following. Let , , and . Then any matrix in can be represented by . By the linearity of . We…

For the orthogonal set , we can define the Lie algebra Note that iff . First note that And since if , . Therefore . Now for the opposite direction, since , then Let . Remark: is topologically closed. That is, it contains its limit. Define as a mapping from any matrix to . Then…