Can you give an example?

sfpotter · 2025-03-30T06:40:05 1743316805

If you mean for how to use Taylor expansions and linear algebra, here's one I just made up.

Let's say I want to differentiate tr(X^T X), tr is the trace, X is a matrix, and X^T is its transpose. Expand:

    tr((X + dX)^T (X + dX)) = tr(X^T X) + 2 tr(X^T dX) + tr(dX^T dX).

Our knowledge of linear algebra tells us that tr is a linear map. Hence, dX -> 2 tr(X^T dX) is the linear mapping corresponding to the Jacobian of tr(X^T X). With a little more work we could figure out how to write it as a matrix.

godelski · 2025-03-30T06:21:35 1743315695

  https://math.stackexchange.com/questions/3680708/what-is-the-difference-between-the-jacobian-hessian-and-the-gradient

  https://carmencincotti.com/2022-08-15/the-jacobian-vs-the-hessian-vs-the-gradient/

vismit2000 · 2025-03-30T05:59:59 1743314399

Check out this classic from 3b1b - How (and why) to raise e to the power of a matrix: https://youtu.be/O85OWBJ2ayo

kgwgk · 2025-03-30T10:33:51 1743330831

For those who prefer reading (I’ve not seen the video, but it seems related):

https://sassafras13.github.io/MatrixExps/

“Thanks to a fabulous video by 3Blue1Brown [1], I am going to present some of the basic concepts behind matrix exponentials and why they are useful in robotics when we are writing down the kinematics and dynamics of a robot.”

esafak · 2025-03-30T15:41:13 1743349273

They didn't show how to actually do it using matrix decomposition!