  Hadamard matrices in $\{0,1\}$ presentation are square $m\times m$ matrices
whose entries are zeros and ones and whose rows considered as vectors in $\Bbb
R^m$ produce the Gram matrix of a special form with respect to the standard
scalar product in $\Bbb R^m$. The concept of Hadamard matrices is extended in
the present paper. As a result pseudo-Hadamard matrices of the first generation
are defined and investigated. An algorithm for generating these pseudo-Hadamard
matrices is designed and is used for testing some conjectures.
