What is a matrix $A_{ij} = i\cdot j$ called?
Let $v = (0, 1, 2, 3, \ldots, \infty)$ and $Tr_n(v) = (0, 1, \ldots, n)$.
Then
$$Tr_n(v)^T Tr_m(v) = = \begin{bmatrix} 0 & 0 & 0 & \dots & 0 & 0 \\ 0 & 1\cdot1 & 1\cdot2 & \dots & 1\cdot(m-1) & 1\cdot(m) \\ 0 & 2\cdot1 & 2\cdot2 & \dots & 2\cdot(m-1) & 2\cdot(m) \\ \vdots & \vdots &\vdots &\vdots &\vdots &\vdots \\ 0 & (n-1)\cdot1 & (n-1)\cdot2 & \dots & (n-1)\cdot(m-1) & (n-1)\cdot(m) \\ 0 & n\cdot1 & n\cdot2 & \dots & n\cdot(m-1) & n\cdot(m) \\ \end{bmatrix} $$
For $m = n$ this matrix is a diagonalizable symmetric matrix which surely has a name?