  We study a discrete-time Markov process on triangular arrays of matrices of
size $d\geq 1$, driven by inverse Wishart random matrices. The components of
the right edge evolve as multiplicative random walks on positive definite
matrices with one-sided interactions and can be viewed as a $d$-dimensional
generalisation of log-gamma polymer partition functions. We establish
intertwining relations to prove that, for suitable initial configurations of
the triangular process, the bottom edge has an autonomous Markovian evolution
with an explicit transition kernel. We then show that, for a special singular
initial configuration, the fixed-time law of the bottom edge is a matrix
Whittaker measure, which we define. To achieve this, we perform a Laplace
approximation that requires solving a constrained minimisation problem for
certain energy functions of matrix arguments on directed graphs.
