Non-square tensors?

Question

I learnt tensor algebra for physics and I never saw a non-square (or non-cubic...) tensor. But, from a mathematical point of view, do non-square tensors exist? And if so, are they used in some area in physics?

Answer 1

I don't know much about the physical uses of a non-square tensor, but there is no mathematical reason for them not to exist.

If we have two vector spaces $V$ and $W$, with dimensions $n$ and $m$ and bases $v_1,v_2,\dots , v_n$ and $w_1, w_2, \dots , w_m$, then the tensor product $V \otimes W$ has basis $v_1 \otimes w_1, v_1 \otimes w_2, \dots, v_1\otimes w_m, \dots, v_n\otimes w_m$, and so is of dimension $nm$. Usually, when writing the components of tensor, we write them in an $m\times n$ grid. So, if $V$ and $W$ are of different dimensions, we will have non-square tensors.

The reason that you see square tensors most often, however, is that we most often care about tensor powers of a given vector space, rather than more general tensor products.