During the course of physics research on nonequilibirum statistical mechanics involving the theory of majorization, I have come across a linear transformation on a d-dimensional vector space that I suspect has been explored by mathematicians, but which is unfamiliar to me.
The transformation is the matrix whose columns are: (1,0,…), (1/2,1/2,0,…), (1/3,1/3,1/3,0,…),…, (1/d,1/d,…,1/d). The In other words:
$$
\left[\begin{array}{cccc}1 & 1/2 & 1/3 & 1/4 & \ldots \\ 0 & 1/2 & 1/3 & 1/4 & \ldots \\ 0 & 0 & 1/3 & 1/4 & \ldots \\ 0 & 0 & 0 & 1/4 & \ldots \\ \ldots & \ldots & \ldots & \ldots & \ldots \end{array}\right]
$$
The inverse transformation is the matrix that has the sequence {1,2,3,…,d} along the main diagonal, and {-1,-2,-3,…,-(d-1)} along the diagonal above the main diagonal. In other words
$$
\left[ \begin{array}{ccccc} 1 & -1 & 0 & 0 & \ldots \\ 0 & 2 & -2 & 0 & \ldots \\ 0 & 0 & 3 & -3 & \ldots \\ 0 & 0 & 0 & 4 & \ldots \\ \ldots & \ldots & \ldots & \ldots & \ldots \end{array}\right]
$$ I would be keen to hear from anyone who has encountered this transformation before and can point me to any relevant literature. Thanks!
