Distance from projections To nilpotents
MacDonald, Gordon W.
Canadian Journal of Mathematics - Journal Canadien de Mathematiques
Mathematical and Computational Sciences
The distance from an arbitrary rank-one projection to the set of nilpotent operators, in the space of k x k matrices with the usual operator norm, is shown to be sec(pi/(k+2))/2. This gives improved bounds for the distance between the set of all non-zero projections and the set of nilpotents in the space of k x k matrices. Another result of note is that the shortest distance between the set of non-zero projections and the set of nilpotents in the space of 3 x 3 matrices is root(3-root 5)/2.