Coordinate-free Definition
We can identify the space of linear operators on a vector space with the space, where . We also have a canonical bilinear function that consists of applying an element of to an element of to get an element of, in symbols . This induces a linear function on the tensor product (by its universal property) which, as it turns out, when that tensor product is viewed as the space of operators, is equal to the trace.
This also clarifies why and why, as composition of operators (multiplication of matrices) and trace can be interpreted as the same pairing. Viewing, one may interpret the composition map as
coming from the pairing on the middle terms. Taking the trace of the product then comes from pairing on the outer terms, while taking the product in the opposite order and then taking the trace just switches which pairing is applied first. On the other hand, taking the trace of and the trace of corresponds to applying the pairing on the left terms and on the right terms (rather than on inner and outer), and is thus different.
In coordinates, this corresponds to indexes: multiplication is given by, so and which is the same, while, which is different.
For finite-dimensional, with basis and dual basis, then is the entry of the matrix of the operator with respect to that basis. Any operator is therefore a sum of the form . With defined as above, . The latter, however, is just the Kronecker delta, being 1 if i=j and 0 otherwise. This shows that is simply the sum of the coefficients along the diagonal. This method, however, makes coordinate invariance an immediate consequence of the definition.
Read more about this topic: Trace (linear Algebra)
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