In this answer to MO question "Geometric interpretation of Trace" (the 9th highest upvoted question on the site!), the following interpretation of the trace is given:
$$\operatorname{Tr}(A) = n\int_{x\in B} \langle Ax, x\rangle \,dm(x)$$ where $B$ is the Euclidean unit sphere ($\mathbb S^{n-1} \subseteq \mathbb R^n$), and $m$ is the uniform measure on $B$ normalised to have total mass $1$.
In other words, $\frac 1n \text{Tr}(A)$ is the average value of the quadratic form $q_A(x)= x^\top A x$ on $\mathbb S^{n-1}$. By coloring the real line $\mathbb R$ with a certain coloring, we can pull back that coloring via $q_A:\mathbb S^{n-1} \to \mathbb R$ to get a coloring of the sphere $\mathbb S^{n-1}$. One can see pictures like this if one searches for spherical harmonics online, e.g. in https://www.chebfun.org/examples/sphere/SphereHeatConduction.html
One can also visualize it as follows (Wiki):
Visual representations of the first few real spherical harmonics. Blue portions represent regions where the function is positive, and yellow portions represent where it is negative. The distance of the surface from the origin indicates the absolute value of ${\displaystyle Y_{\ell }^{m}(\theta ,\varphi )}$ in angular direction $(\theta,\varphi)$.
I did this for the quadratic form $q_A(x)$ (i.e. I graphed the function $[x\mapsto q_A(x)x]: \mathbb S^{n-1}\to \mathbb R^n$ in Desmos, in both 2D: https://www.desmos.com/calculator/wounhuhlzm and 3D: https://www.desmos.com/3d/d34fdc19f8.
I'm curious about the following:
Question: can we derive an expression for how the initial condition $u(x,t=0) = q_A(x)$ on the sphere evolves under the heat/diffusion equation? Intuitively, it should end up at the average state $u(x,t=\infty) = \frac 1n \text{Tr}(A)$. Perhaps, there are a sequence of matrices $A_t$ s.t. $A_0=A$ and $A_\infty = \text{diag}(\frac 1n \text{Tr}(A))$ so that $u(x,t) = x^\top A_t x$.
Unfortunately I don't even quite know how to start, as I don't even really know the heat equation on spheres. https://en.wikipedia.org/wiki/Laplace%E2%80%93Beltrami_operator#Spherical_Laplacian gives an expression for the Laplacian (Laplace-Beltrami operator), but even if this gives the correct heat equation on the sphere, I wouldn't know how to begin with the question above.