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,

# Category: math

## Rodrigues’ rotation formula

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…

## ARIMA

It is ARMA after applying lap-1 and lap-M difference to data.

## Formal definition of Lie algebra

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…

## Lie algebra of sl(2)

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…

## Lie algebra of O(n)

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…

## Lie group and Lie algebra

Found a very nice series on Lie group and Lie algebra. Watched the first couple lectures. The main message I got so far is that Lie algebra combine techniques from analysis (calculus) with algebra (group). And one nice thing I learned is that $latex \exp(A+B) = \exp(A) \exp(B)$ if $latex AB = BA$. I didn’t…

## Weiszfeld’s algorithm

Came across Weiszfeld’s algorithm while reading the quaternion equivariant capsule paper. Weiszfeld’s algorithm is just a form of iteratively reweighted least square. And the algorithm is well illustrated with this tweet.

## Jordan normal form

Any square matrix can be decomposed into Jordan normal form. The Jordan block can be decomposed into different numbers of Jordan boxes depending on the geometric multiplicity of the corresponding eigenvalues (only one box if the corresponding geometric multiplicity is one). This video series gives an excellent description to compute the Jordan normal form.

## Robust PCA

Came across of this video explaining robust PCA. It came with a book and it looks okay. As PCA can be considered as decomposition of data matrix with the highest singular value. So what PCA is doing is simply low-rank matrix approximation of the data matrix. So robust PCA is a simple idea that tries…