Creation and annihilation in matrix theory

A closed form representation is given for the matrix S that sweeps out a si ngle column. This includes the integer as well as the unitary and regular c ases. The closed form is built up from the 2 x 2 case. The product rule for adjoints is used to show that the dual of this procedure is precisely the completion procedure, which completes a single column to a matrix B such th at SB = BS = diagonal. This construction underlines the fact that the compl etion and elimination processes are complementary. By permuting rows suitab ly, the matrices may be assumed to be in lower Hessenberg form. (C) 2000 Pu blished by Elsevier Science Inc. All rights reserved.