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....Read more