Geodesic
Binary Moore-Penrose Inverses of Set Inclusion Incidence Matrices
Séminaire lotharingien de combinatoire, Tome 45 (2000-2001)

Voir la notice de l'acte provenant de la source Séminaire Lotharingien de Combinatoire website

This note is a supplement to some recent work of R. B. Bapat on Moore-Penrose inverses of set inclusion matrices. Among other things Bapat constructs these inverses (in case of existence) for H(s,k) mod p, p an arbitrary prime, 0 = s = k = v-s. Here we restrict ourselves to p=2. We give conditions for s,k which are easy to state and which ensure that the Moore-Penrose inverse of H(s,k) mod 2 equals its transpose. E.g., H(s,v-s) mod 2 has this property. Furthermore Ker H(s,v-s) mod 2 is nonzero if 0 2s v = 3s and then there is a decomposition

$\displaystyle {\rm Ker}\, H(s,v-s) \equiv \underset {2 \mid \binom{v-s-j } { v-2s}} {\sum _ {0 \le j \le s-1}} {\rm Im}\,H(v-s,v-j)\ {\rm mod}\ 2.$


Also, refinements of this decomposition are given. 

@article{SLC_2000-2001_45_a3,
     author = {Helmut Kr\"amer},
     title = {Binary {Moore-Penrose} {Inverses} of {Set} {Inclusion} {Incidence} {Matrices}},
     journal = {S\'eminaire lotharingien de combinatoire},
     publisher = {mathdoc},
     volume = {45},
     year = {2000-2001},
     url = {http://geodesic.mathdoc.fr/item/SLC_2000-2001_45_a3/}
}
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Helmut Krämer. Binary Moore-Penrose Inverses of Set Inclusion Incidence Matrices. Séminaire lotharingien de combinatoire, Tome 45 (2000-2001). http://geodesic.mathdoc.fr/item/SLC_2000-2001_45_a3/