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 operator, matrix, nilpotent, idempotent, projection

MSC Classifications:

47A15, 47D03, 15A30 show english descriptions Invariant subspaces [See also 47A46]
Groups and semigroups of linear operators {For nonlinear operators, see 47H20; see also 20M20}
Algebraic systems of matrices [See also 16S50, 20Gxx, 20Hxx] 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]

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