ELA, Volume 10, pp. 190-200, July 2003, abstract.

Absolutely Flat Idempotents

Charles R. Johnson, Yonatan Harel, Christopher J. Hillar,
Jonathan M. Groves, and Patrick X. Rault

A real n-by-n idempotent matrix A with all entries having
the same absolute value is called absolutely flat. The 
possible ranks of such matrices are considered along with
a characterization of the triples: size, constant, and rank 
for which such a matrix exists. Possible inequivalent examples
of such matrices are also discussed.