In my work I have come about two type of matrices:
1-Let $U_{np}\in \mathbb{R}^{n\times p}$ be such that $(U_{np})_{ij}=p-j+1$, e.g. $U_{53}=\begin{bmatrix}3 & 2 & 1\\ 3 & 2 & 1\\ 3 & 2 & 1\\ 3 & 2 & 1\\ 3 & 2 & 1 \end{bmatrix}.$
2-Let $V_p\in\mathbb{R}^{p\times p}$ be such that $(V_p)_{ij}=p-\max(i,j)+1$, e.g. $V_5=\begin{bmatrix}5 & 4 & 3 & 2 & 1\\ 4 & 4 & 3 & 2 & 1\\ 3 & 3 & 3 & 2 & 1\\ 2 & 2 & 2 & 2 & 1\\ 1 & 1 & 1 & 1 & 1 \end{bmatrix}.$
I just want to know if these type of matrices have specific names or if anybody has seen them before, and could provide me with some references.
More context:
I am using the above matrices to simplify the following expressions:
$$\begin{align*} \sum_{i=1}^{p}AI_{i;p} & =U_{np}\odot A,\text{ and }\\ \sum_{i=1}^{p}I_{i;p}A'AI_{i;p} & =V_{p}\odot A'A, \end{align*},$$ where, $A\in\mathbb{R}^{n\times p}$, $\odot$ is the is the Hadamard (element-wise) product, and $I_{i;p}$ is an $p\times p$ matrix with all elements zero except the first $i$ diagonal terms or equivalently an $p\times p$ identity matrix where the last $p-i$ diagonal elements are zero. For example $I_{2;3}=\begin{bmatrix}1 & 0 & 0\\ 0 & 1 & 0\\ 0 & 0 & 0 \end{bmatrix}.$
The first matrix is a rank-one matrix product of the column vector with 5 ones and the row vector $(1 \ 2 \ 3)$. I don't see what can be said more...
The second matrix is characterized by its inverse. This inverse is (one form of) the (tridiagonal) discrete second differentiation matrix with entries $-1, \ 2, \ -1$ (with an exception at the beginning). See for example slide $11$ of {http://www.mathematik.uni-dortmund.de/~kuzmin/cfdintro/lecture4.pdf}
You mention a coupling between these matrices. I don't see any...