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# Distance from Idempotents to Nilpotents

Published:2007-06-01
Printed: Jun 2007
• Gordon W. MacDonald
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## Abstract

We give bounds on the distance from a non-zero idempotent to the set of nilpotents in the set of $n\times n$ matrices in terms of the norm of the idempotent. We construct explicit idempotents and nilpotents which achieve these distances, and determine exact distances in some special cases.
 Keywords: operator, matrix, nilpotent, idempotent, projection
 MSC Classifications: 47A15 - Invariant subspaces [See also 47A46] 47D03 - Groups and semigroups of linear operators {For nonlinear operators, see 47H20; see also 20M20} 15A30 - Algebraic systems of matrices [See also 16S50, 20Gxx, 20Hxx]