Moore-Penrose inverse of set inclusion matrices (Q1590644)

The set inclusion matrix \(W_{sk}\) with dimensions \(\binom vs\times\binom vk\), where \(s\), \(k\) and \(v\) are integers satisfying \(0\leq s\leq k\leq v\), is defined as follows. The rows and columns of \(W_{sk}\) are indexed by the \(s\)-element subsets and by the \(k\)-element subsets of a \(v\)-element set, respectively, and the entry in row \(S\) and column \(U\) of \(W_{sk}\) is \(1\) if \(S \subset U\) and \(0\) otherwise. Defining the Moore-Penrose inverse of an \(m \times n\) matrix \(A\) as an \(n \times m\) matrix \(G\) satisfying the equations \(AGA = A\), \(GAG = G\), \((AG)^T = AG\) and \((GA)^T = GA\), an explicit formula for the Moore-Penrose inverse of \(W_{sk}\) over the set of reals is given. The eigenvalues of \(W_{sk}W_{sk}^T\) are also determined. The main results are a necessary and sufficient condition for \(W_{sk}\) to admit a Moore-Penrose inverse over the set of integers modulo a prime \(p\) and the formula for the Moore-Penrose inverse if it exists. An example with \(s = 2\), \(k = 4\), \(v = 6\) and \(p = 3\) is discussed.