And interestingly, both are connected. If d_i somewhat hand-wavingly expresses the vector d_i = (0, ..., 0, 1, 0, ... 0) with the 1 at position i, then given matrix M you can do

f(i, j) := d_i^T * M * d_j

The RHS is using classical matrix multiplication, and the function value will be the matrix' entry at column i, row j.