The setting
Consider a family of matrices of dimensions dimensions $2^d \times 2^d$, with entries:
$$ A_{ij} = \begin{cases} 1 & \text{if}~(i ~\land~ j) = i\\ 0 & \text{otherwise} \\ \end{cases} $$
where $(\land)$ is bitwise-AND.
That is, if we interpret $i$ and $j$ as the characteristic vectors for subsets subset of size $d$, then:
$$A_{ij} = 1 \iff i \subseteq j$$
An example
For example, in the 2-D case:
2x2
---
⎡1 1⎤ // emptyset in {*} | {*} in {*}
⎢ ⎥
⎣0 1⎦ //emptyset in emptyset, | {*} notin emptyset
This matrix corresponds to subsets of the set $\{ * \}$.
the row $[1, 1]$ corresponds to the subset $\{ * \}$, since $\{ * \} \subseteq \{ * \}$, $\emptyset \subseteq \{ * \}$,
the row $[0, 1]$ correspond to the subset $\emptyset$, since $\emptyset \not \subseteq \{ * \}$, $\emptyset \subseteq \emptyset$, .
The question
What is the name for this matrix? Experiments seem to indicate that it has all $1$ eigenvalues, is full-rank, non-diagonalizable.
Do its eigenvalues mean something?
Experiments
Experimentally, here are the first few:
4x4
----
⎡1 1 1 1⎤
⎢ ⎥
⎢0 1 0 1⎥
⎢ ⎥
⎢0 0 1 1⎥
⎢ ⎥
⎣0 0 0 1⎦
8x8
---
⎡1 1 1 1 1 1 1 1⎤
⎢ ⎥
⎢0 1 0 1 0 1 0 1⎥
⎢ ⎥
⎢0 0 1 1 0 0 1 1⎥
⎢ ⎥
⎢0 0 0 1 0 0 0 1⎥
⎢ ⎥
⎢0 0 0 0 1 1 1 1⎥
⎢ ⎥
⎢0 0 0 0 0 1 0 1⎥
⎢ ⎥
⎢0 0 0 0 0 0 1 1⎥
⎢ ⎥
⎣0 0 0 0 0 0 0 1⎦