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I was trying to think about a geometric interpretation of the trace of a matrix. I have kind of an intuition about what it is, thinking about it as the sum of the eigenvalues and relating this with the concept of divergence. I understand the trace as a number that measures "more or less" how a linear map "in average" scales all the vectors of the space. However, I'm not able to find a more satisfactory and precise geometric interpretation of it (just this explanation, which is kind of imprecise) and, what's more, I'm not able to fully understand why the trace remains invariant by changes of basis (I understand the algebraic proof, but I don't get an intuition of what is happening).

I'm familiar with the geometric interpretation of the determinant, as the scaling factor of the area of a unit square, volume of the unit cube... and, for that reason, I can fully understand why the determinant remains invariant by changes of basis. I was wondering if there is something similar for the interpretation of the trace and I am missing it.

Thank you for your help:)

Cristina Lopez
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  • There are several thoughts on this issue on the "rival to MSE" website Quora; https://www.quora.com/What-is-a-mathematically-intuitive-way-to-understand-the-trace-of-a-matrix I guess the gauntlet now thrown down is can a better answer be given here (Ha, ha!) – Martin Hansen Dec 29 '21 at 11:51
  • There are some outstanding answers on the sister website Maths Overflow; https://mathoverflow.net/questions/13526/geometric-interpretation-of-trace – Martin Hansen Dec 29 '21 at 12:02

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