The Trace of a Matrix: A Dual and Tensor Perspective

The main result

Dear Reader, in today's brief, we will discuss the trace of a square matrix. Let us try to have a slightly different perspective on this familiar object, which takes a square matrix (thus, a linear operator) and returns a number of the same type as matrix elements. First, one can notice that this function is linear. Hence, it belongs to the dual vector space of linear operators. It turns out that the trace has a pretty unique characterization in terms of the tensor product operation. Moreover, the trace allows us to glorify the space of linear transformations to the inner product space. On our way, we define properly the transpose of a linear operator, which must pass by the dual, a fact too often omitted in standard linear algebra courses. Some details of this brief story can be found in the slide below for your reflection, meditation, and enjoyment.