The real meaning of trace of matrix | Lie groups, algebras, brackets #5

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. Files for download: Go to and enter the following password: traceisdiv 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: 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): (linear_algebra)#Derivative_relationships Divergence (more qualitative, and subtly different from the video): #Physical_interpretation_of_divergence Jacobi’s formula (more formal proof): ’s_formula Other than commenting on the video, you are very welcome to fill in a Google form linked below, which helps me make better videos by catering for your math levels: If you want to know more interesting Mathematics, stay tuned for the next video! SUBSCRIBE and see you in the next video! If you are wondering how I made all these videos, even though it is stylistically similar to 3Blue1Brown, I don’t use his animation engine Manim, but I use PowerPoint, GeoGebra, and (sometimes) Mathematica to produce the videos. Social media: Facebook: Instagram: Twitter: Patreon: (support if you want to and can afford to!) Merch: Ko-fi: [for one-time support] For my contact email, check my About page on a PC. See you next time!