  This article extends the tensor network Kalman filter to matrix outputs with
an application in recursive identification of discrete-time nonlinear
multiple-input-multiple-output (MIMO) Volterra systems. This extension
completely supersedes previous work, where only $l$ scalar outputs were
considered. The Kalman tensor equations are modified to accommodate for matrix
outputs and their implementation using tensor networks is discussed. The MIMO
Volterra system identification application requires the conversion of the
output model matrix with a row-wise Kronecker product structure into its
corresponding tensor network, for which we propose an efficient algorithm.
Numerical experiments demonstrate both the efficacy of the proposed matrix
conversion algorithm and the improved convergence of the Volterra kernel
estimates when using matrix outputs.
