Abstract
Let A be an n × n real matrix. A is called {0,1}-cp if it can be factorized as A = BB T with bij =0 or 1. The smallest possible number of columns of B in such a factorization is called the {0,1}-rank of A. A {0,1}-cp matrix A is called minimal if for every nonzero nonnegative n × n diagonal matrix D, A-D is not {0,1}-cp, and r-uniform if it can be factorized as A=BB T, where B is a (0, 1) matrix with r 1s in each column. In this article, we first present a necessary condition for a nonsingular matrix to be {0,1}-cp. Then we characterize r-uniform {0,1}-cp matrices. We also obtain some necessary conditions and sufficient conditions for a matrix to be minimal {0,1}-cp, and present some bounds for {0,1}-ranks.
Acknowledgement
A. Berman was supported by the New York Metropolitan Research Fund at the Technion. C. Xu was supported by a postdoctoral scholarship from the Israel Council for Higher education. We are grateful to the referee for his/her helpful comments.