Moore-Penrose inverse of set inclusion matrices
type: Item
instance of: scholarly article
title: Moore-Penrose inverse of set inclusion matrices
zbMATH Open document ID: 0963.15004
DOI: 10.1016/S0024-3795(00)00123-3
published in: Linear Algebra and its Applications
publication date: 2001-06-18T00:00:00Z
review text: 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.
Mathematics Subject Classification ID: 15A09
Mathematics Subject Classification ID: 15B57
Mathematics Subject Classification ID: 15A18
zbMATH DE Number: 1547886
zbMATH Keywords: set inclusion matrix
zbMATH Keywords: generalized inverse
zbMATH Keywords: Moore-Penrose inverse
zbMATH Keywords: incidence matrix
zbMATH Keywords: finite field
zbMATH Keywords: eigenvalues
author: R. B. Bapat
MaRDI profile type: Publication
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Wikidata QID: Q127549223
description: scientific article; zbMATH DE number 1547886