What's the intuitive meaning of matrix trace, if any?

Question

3Blue1Brown has great visual explanations of what some commonly used properties of matrices mean, such as determinant, rank, and kernel, but I have absolutely no idea what the trace of a matrix has in linear algebra and what its visual representation (if any) is.

Answer 1

An example is the Frobenius norm of an operator in a square matrix representation. The trace does not depend on the base of the matrix representation (is invariant). If $A$ is the matrix (or the linear operator), then $\|A\| = \sqrt{\mathrm{tr}(A^* A)}$. And The "operator norm of a linear operator $T:V->W$ is the largest value by which $T$ stretches an element of $V$". There are also, of course, other examples or interpretations of the trace of a matrix.

Answer 2

The trace of the infinitesimal strain tensor, $\boldsymbol{\varepsilon},$ in the theory of elasticity represents a fractional volume change in the elastic material when it is deformed, so that $$\dfrac{V_{\rm deformed}-V_{\rm initial}}{V_{\rm initial}}= \varepsilon_{11}+\varepsilon_{22}+\varepsilon_{33}, $$ and minus one third of the trace of the Cauchy stress tensor, $\boldsymbol{\sigma},$ represents the isotropic compressive pressure in the material, $$p=-\dfrac{\sigma_{11}+\sigma_{22}+\sigma_{33}}{3}.$$