Part 6: https://youtu.be/gj4kvpy1eCE
Can we visualise this algebraic procedure of adding diagonal entries? What is really happening when we add them together? By visualising it, it is possible to almost immediately see how the different properties of trace comes about.
This channel is meant to showcase interesting but underrated maths (and physics) topics and approaches, either with completely novel topics, or a well-known topic with a novel approach. If the novel approach resonates better with you, great! But the videos have never meant to be pedagogical - in fact, please please PLEASE do NOT use YouTube videos to learn a subject.
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The concept of the whole video starts from one line the Wikipedia page about trace, and I am surprised this isn't on YouTube: "A related characterization of the trace applies to linear vector fields. Given a matrix A, define a vector field F on R^n by F(x) = Ax. The components of this vector field are linear functions (given by the rows of A). Its divergence div F is a constant function, whose value is equal to tr(A)."
Actually, this is one of the last concepts in linear algebra that I really wanted a visualisation for, the other being transpose, but this is already on the channel: https://www.youtube.com/watch?v=g4ecBFmvAYU
Chapters:
00:00 Introduction
00:48 Matrix as vector field
02:24 Divergence
04:50 Connection between trace and divergence
10:12 Trace = sum of eigenvalues
13:32 Determinant and matrix exponentials
15:15 Trace is basis-independent
18:10 Jacobi's formula
Further reading:
Trace (the origin of the whole video): https://en.wikipedia.org/wiki/Trace_(linear_algebra)#Derivative_relationships
Divergence (more qualitative, and subtly different from the video): https://en.wikipedia.org/wiki/Divergence#Physical_interpretation_of_divergence
Jacobi's formula (more formal proof): https://en.wikipedia.org/wiki/Jacobi%27s_formula
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