I have a linear time-invariant system with $m$ inputs and $n$ outputs in the form
\begin{equation} \textbf Y(z) = \textbf U(z)\textbf G(z), \end{equation}
where $\bf Y$$(z)$ is the output vector of shape $1{\times}n$, $\bf U$$(z)$ is the input vector of shape $1{\times}m$ and $\bf G$$(z)$ is a matrix of transfer functions of shape $m{\times}n$. Each element of $\bf G$$(z)$ is a transfer function from each input to each output.
How should this be written in order to express the transfer function matrix on the RHS?
In the single channel case I would simply write
\begin{equation} H(z)=\frac{Y(z)}{U(z)} = G(z), \end{equation}
but I'm assuming this ratio doesn't make sense when using matrices since a) $\bf U$$(z)$ is non-invertible, and b) even if it were invertible, the operand dimensions ($1{\times}n$ and $1{\times}m$) could not be multiplied. Is there an equivalent notation for matrices?
Apologies for the simple question, I've not dealt with matrices in much depth before! Any recommendations for reading would also be appreciated.