9
$\begingroup$

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?

$\endgroup$
14
$\begingroup$

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.

$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.