Is there a name for matrices of this form?
$T\in \mathbb{R}^{m\times n}, T_{i,j}= \begin{array}{cc} \{ \begin{array}{cc} 1 & i=j \\ 0 & \text{True} \\ \end{array} \\ \end{array}$
Such as these:
$\begin{array}{cccc} \left( \begin{array}{c} 1 \\ \end{array} \right) & \left( \begin{array}{cc} 1 & 0 \\ \end{array} \right) & \left( \begin{array}{ccc} 1 & 0 & 0 \\ \end{array} \right) & \left( \begin{array}{cccc} 1 & 0 & 0 & 0 \\ \end{array} \right) \\ \left( \begin{array}{c} 1 \\ 0 \\ \end{array} \right) & \left( \begin{array}{cc} 1 & 0 \\ 0 & 1 \\ \end{array} \right) & \left( \begin{array}{ccc} 1 & 0 & 0 \\ 0 & 1 & 0 \\ \end{array} \right) & \left( \begin{array}{cccc} 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ \end{array} \right) \\ \left( \begin{array}{c} 1 \\ 0 \\ 0 \\ \end{array} \right) & \left( \begin{array}{cc} 1 & 0 \\ 0 & 1 \\ 0 & 0 \\ \end{array} \right) & \left( \begin{array}{ccc} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \\ \end{array} \right) & \left( \begin{array}{cccc} 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ \end{array} \right) \\ \left( \begin{array}{c} 1 \\ 0 \\ 0 \\ 0 \\ \end{array} \right) & \left( \begin{array}{cc} 1 & 0 \\ 0 & 1 \\ 0 & 0 \\ 0 & 0 \\ \end{array} \right) & \left( \begin{array}{ccc} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \\ 0 & 0 & 0 \\ \end{array} \right) & \left( \begin{array}{cccc} 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \\ \end{array} \right) \\ \end{array}$