The geometric meaning of the determinant of a matrix as an area or a volume is dealt with in many textbooks. However, I don't know if the trace of a matrix has a geometric meaning too.
Is there any geometric or physical (intuitive) significance related to the trace of a matrix?
Yes, the trace of a matrix has several geometric and physical interpretations, although they are often more subtle than those of the determinant. Below are key perspectives on the significance of the trace.
$1$. Trace as the Sum of Eigenvalues
For a square matrix $ A \in \mathbb{R}^{n \times n} $, the trace is the sum of its eigenvalues (counted with multiplicities): $$ \operatorname{Tr}(A) = \sum_{i=1}^n \lambda_i. $$ This makes the trace invariant under change of basis. Geometrically, this can be interpreted as a kind of average directional scaling of the transformation $ A $ in its eigenbasis.
$2$. Trace and Divergence in Vector Fields
In vector calculus, the trace of the Jacobian matrix of a vector field corresponds to its divergence. Let $ \mathbf{F} = (F_1, \ldots, F_n) $ be a vector field on $ \mathbb{R}^n $. The divergence is $$ \operatorname{div} \mathbf{F} = \sum_{i=1}^n \frac{\partial F_i}{\partial x_i} = \operatorname{Tr}(J(\mathbf{F})). $$ Geometric meaning: The divergence at a point measures whether the vector field is expanding (source) or contracting (sink) at that point.
$3$. Trace as a Linear Invariant
Given a linear transformation $ T: V \to V$, its trace is independent of the basis chosen. It serves as a useful invariant in many contexts:
$a)$ In the theory of linear operators,
$b)$ In representation theory (character theory),
$c)$ In quantum mechanics, where $ \operatorname{Tr}(\rho A) $ computes the expected value of an observable $ A $ with respect to a density operator $ \rho $.
$4$. Trace and Volume Change (Infinitesimally)}
While the determinant gives the global volume scaling of a linear transformation, the trace appears in the exponential map: $$ \det(e^{tA}) = e^{t \cdot \operatorname{Tr}(A)}. $$ This connects the trace to the infinitesimal rate of volume change under continuous-time flows like those found in dynamical systems.
