In both string diagrams (i.e. Penrose graphical notation) and tensor index notation, the trace of a linear map has a nice representation, either as drawing a string connecting a tensor's output back to its own input, or as summing over the same tensor's covariant and contravariant indices (which is a sort of application of an object to itself). This makes some properties of the trace obvious, such as invariance under cyclic permutations (which you can think of as moving beads on a string, or as renaming indices) and the fact that the trace of a tensor product produces composition. But I'm not sure what this notation should indicate about what a trace is.
For most traced monoidal categories, this notation makes a lot of intuitive sense, as expressing sort of fixed point. But in the case of vector spaces, other than eigenvalues being generalized fixed points and the trace being their sum, I don't really know how to interpret it (this might have to do with the fact that this category has nontrivial scalars, unlike most other monoidal categories I come across).
Is there a nice way to think about traces from this perspective?
A vector space $\;V\;$ with a dot product $\;(v,w)\to v\cdot w\;$ and an orthonomal basis $\;(e_1,e_2,\dots,e_n)\;$ with $\;\delta_{i,j}=e_i\cdot e_j\;\;\forall i,j\;$ yields to two linear maps. $\;L_i:R\to V\;$ defined by $\;L_i(x) := x\; e_i\;$ and $\;L_i^*:V\to R\;$ defined by $\;L_i^*(v)=v\cdot e_i.\;$ Any linear map $\;T:V\to V\;$ is uniquely determined by where basis elements go. Thus $\;T=\sum_{i,j}a_{i,j}L_{i,j}\;$ where the $\;a_{i,j}\;$ are the matrix entries of $\;T\;$ and $\;L_{i,j}(v) := L_iL_j^*(v).\;$ By definition, $\;\textrm{trace}(T):=\sum_ia_{i,i}.\;$ This can be interpreted as for each $i$ the basis vector $\;e_i\;$ is mapped out by $\;T\;$ to $\;T(e_i)\;$ and the $\;i$-th component of that is input as $\;a_{i,i}$ and the trace is the sum of those contributions.