  We study positive definite functions on noncommuting strict contractions. In
particular, we study functions that induce positive definite Hua-Bellman
matrices (i.e., matrices of the form $[\det(I-A_i^*A_j)^{-\alpha}]_{ij}$ where
$A_i$ and $A_j$ are strict contractions and $\alpha\in\mathbb{C}$). We start by
revisiting a 1959 work of \citeauthor{bellman1959}
(R.~Bellman~\emph{Representation theorems and inequalities for Hermitian
matrices}; Duke Mathematical J., 26(3), 1959) that studies Hua-Bellman matrices
and claims a strengthening of \citeauthor{hua1955}'s representation theoretic
results on their positive definiteness~(L.-K. Hua, \emph{Inequalities involving
determinants}; Acta Mathematica Sinica, 5(1955), pp.~463--470). We uncover a
critical error in Bellman's proof that has surprisingly escaped notice to date.
We "fix" this error and provide conditions under which $\det(I-A^*B)^{-\alpha}$
is a positive definite function; our conditions correct Bellman's claim and
subsume both Bellman's and Hua's prior results. Subsequently, we build on our
result and introduce a new hyperbolic-like geometry on noncommuting
contractions, and remark on its potential applications.
