  The Symmetric Nonnegative Matrix Trifactorization (SN-Trifactorization) is a
factorization of an $n \times n$ nonnegative symmetric matrix $A$ of the form
$BCB^T$, where $C$ is a $k \times k$ symmetric matrix, and both $B$ and $C$ are
required to be nonnegative. This work introduces the SNT-rank of $A$, as the
minimal $k$, for which such factorization exists. After listing basic
properties and exploring SNT-rank of low rank matrices, the class of
nonnegative symmetric matrices with SNT-rank equal to rank is studied. The
paper concludes with a completion problem, that asks for matrices with the
smallest possible SNT-rank among all nonnegative symmetric matrices with given
diagonal blocks.
