Formal definition of Lie algebra

Lie algebra is a vector space with a map such that is bilinear Jacobi inequality: . Note that 2) since . Note that the converse is true most of time as well since that implies . As long as we can divide by 2 (in the field where ), we have .    

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 can…

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 if . I didn’t realize the commutation condition is necessary earlier. When…

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…

Banach fixed point theorem

I came across the proof of Perron Frobenius theorem here and it used Banach fixed point theorem. So I spent some time to understand the fixed point theorem from the wiki. Theorem Banach fixed point theorem considers a complete metric space (with metric ) and for any contraction mapping , there will exist a unique…

Levy process

IMHO, the Levy process is just a generalization of the Poisson process. For example, consider as the number of arrivals by time as in the Poisson process. We expect the followings will be satisfied for Random variables and are independent if and do not overlap for some distribution . Meaning that probability distribution only depends…

Proof of Fourier inversion formula

I think I knew how to show it before. But suddenly just realize I can’t. Say let’s define and the inverse transform . So we expect So one can finish the proof as we can show that  , note that as . “Intuitively”, the above does appear to be a -function. A more rigorous proof…