I would like to understand better all the properties of a a particular matrix type. The problem is that I do not know how these matrices are called.
The matrix I consider is a binary matrix somehow related to permutation matrices and projection matrices. I use them as a way to index a vector (or other matrices). I checked the wiki articles of binary, permutation and projection matrix without finding the name for such a matrix.
Let me use an example to explain it better.
Consider the vector $v = [2\ 1\ 5\ 0\ 4\ 1]^T$ and the following indeces $\mathcal{I} = (1, 3, 4, 6)$. Now consider the linear transformation from $\mathbb{R}^6 \rightarrow\mathbb{R}^4$ described by $\tilde{v} = v(\mathcal{I}) = [2\ 5\ 0\ 1]^T$. The matrix which perform such an operation is obtained from the identity matrix $$A = I_6(\mathcal{I}) = \begin{bmatrix} 1&0&0&0&0&0\\0&0&1&0&0&0\\ 0&0&0&1&0&0\\ 0&0&0&0&0&1 \end{bmatrix}$$ $A$ is like a permutation (a unique one for each row or column) but it is not as it is no square matrix. It is also related to projection because $A^TA$ is a projector. Furthermore $AA^T = I_4$.
Do they belong to a particular matrix typology?
More generally consider indeces with possible repetition like $\mathcal{J}=[3\ 1\ 2\ 2\ 4\ 5\ 1]$. The resulting matrix (as above) $I_6(\mathcal{J})$ have a $1$ on each row but more on some column. Do this kind of matrix fall in some typology as well?