What information does the trace of a matrix give?

I was recently thinking what information do we get from a matrix. So if we say the columns (or rows) of a matrix define the basis of a system, say vectors of 3 dimensional space. Then the determinant will tell about the volume of the space enclosed by those vectors. Does trace give any information about the space?

I) More generally,

• Let $V$ be a (say, finite dimensional) vector space over a field $\mathbb{F}$.

• Let $(e_i)_{i\in I}$ be a basis for $V$.

• Let $A\in {\rm End}(V)$ be an endomorphism in $V$, i.e. a $\mathbb{F}$-linear map $A:V\to V$.

• Let the matrix $(M^i{}_j)_{i,j\in I}$ be the unique $\mathbb{F}$-valued matrix that represents the linear map $A$ in the basis $(e_i)_{i\in I}$, which means that $$\forall j\in I:~~ Ae_j ~=~ \sum_{i\in I} e_iM^i{}_j .$$

Then the trace ${\rm tr(A)}$ of the linear map $A$, defined as $${\rm tr(A)}~:=~\sum_{i\in I}M^i{}_i,$$ does not depend on the choice of basis $(e_i)_{i\in I}$. For this reason it is an important invariant.

II) Another invariant is the determinant $\det (A)$ of $A$. The physical interpretation of the determinant is the (signed) quotient (often called the Jacobian $J$) between the (signed) volume of a region in $V$ before and after applying the map $A$.

III) A physical interpretation of the trace can be provided via i) the physical interpretation of the determinant and ii) the formula

$$\det (e^A) ~=~e^{\rm tr(A)}.$$

Therefore, for an infinitesimal map $A$, $$\det ({\bf 1}+A)~\approx~1+ {\rm tr(A)}.$$ So roughly speaking, in one interpretation, the trace provides information about infinitesimal change in the volume factor.