1 norm $\|\|_1$, of non square matrix

Does $ 1$ norm exist for non-square matrices? By $ 1$ norm I mean

$ d (x,y)=\sum_{i=1}^{n} |x^i-y^i|, x=(x_1,\dots, x_n), y=(y_1,\dots, y_n)$

Suppose $ A$ is $ m\times n, (m\ne n)$ matrix what can we say about $ \|A\|_1$ ? Also, can we say $ \|A\|_1=\|A^T\|_1= \|A^TA\|_1=\|AA^T\|_1$ ? and $ \|AB\|_1\le \|A\|_1\|B\|_1$ Thanks for helping.