Unitary dilation of a rectangular contraction

Question

Given a contraction $A$ (which means $||A||\leqslant 1$) on a Hilbert space $H$, one can show that $A$ has a unitary dilation $U_A$ on a larger Hilbert space $K$, i.e. $A=P_H U_A|_H$. Let's say we're interested only in finite dimensional spaces. If we define a defect operator $D_A=\sqrt{1-A^{\dagger}A}$, an example of a unitary dilation is \begin{equation*} U_A=\left( \begin{array}{cc} A & -D_{A^{\dagger}} \\ D_A & A^{\dagger} \end{array} \right). \end{equation*} Here's the question: one can show that the above dilation is also unitary when $A$ is a rectangular (e.g. $A:\mathbb{C}^n\rightarrow\mathbb{C}^m,n\neq m$) contraction. However, in the literature it is always assumed that $A$ is square-dimensional. What is the reason of this assumption?