# What is meant by "No tensor may be expressed as a single tensor product"?

My book about Nonlinear Solid Mechanics contains the following sentence (also see picture below):

Note further that no tensor may be expressed as a single tensor product, in general, $$\mathbf{A} = \mathbf{u}\otimes\mathbf{v} + \mathbf{w}\otimes\mathbf{x} \neq \mathbf{y}\otimes\mathbf{z}$$.

I can't really make sense of this, I mean, theoretically both $$\mathbf{u}\otimes\mathbf{v} + \mathbf{w}\otimes\mathbf{x}$$ and $$\mathbf{y}\otimes\mathbf{z}$$ should be tensors? What is this sentence trying to tell me?

• My best guess is that they're trying to convey the fact that sums of simple tensors need not be simple tensors (by simple tensor I mean a tensor of the form $a\otimes b$). Aug 16 at 22:10
• It is even worse: some cannot even be expressed as the sum of two of them Aug 16 at 22:15
• (Unless the ordinary vectors you start with live in a two-dimensional space) Aug 16 at 22:15
• That is why the space $U \otimes V$ (for linear spaces $U$ and $V$) has dimension equal to the product of the dimensions of $U$ and $V$ (hence is HUGE compared to the space $U \oplus V$ consisting of tuples $(u, v)$ with $u \in U$ and $v \in V$ and whose dimension is only the sum of the dimensions of $U$ and $V$) Aug 16 at 22:18
• they express it pourly with their 'no tensor': of course some tensors can be expressed in the form $x \otimes y$, for instance the tensor $x \otimes y$ itself... But what they try to convey is what Michael says: that in general not all tensors are simple tensors. The set of simple tensors is a very thin subset of a very weird geometric shape floating around in the much much bigger (and simpler looking) linear space of all tensors which is defined as the set of all linear combinatrions of simple tensors (subject to the rules about which are equal to which) Aug 16 at 22:21

What this means is that if we take two (say real) vector spaces $$U$$ and $$V$$, then in general it is NOT true that for every $$\xi\in U\otimes V$$, there exist $$u\in U$$, and $$v\in V$$ such that $$\xi=u\otimes v$$. Said another way, such pure/simple tensors $$\{u\otimes v\}_{u\in U, v\in V}$$ are special things. Also, in general, the sum of pure tensors need not again be a pure tensor.
On the other hand, every tensor can be expressed as a linear combination of pure tensors. More precisely, if $$\{u_1,\dots, u_n\}$$ and $$\{v_1,\dots, v_m\}$$ are bases for $$U$$ and $$V$$ respectively, then $$\{u_i\otimes v_j\,:\, 1\leq i\leq n, \,1\leq j\leq m\}$$ will be a basis for the space of tensors $$U\otimes V$$, so for every $$\xi\in U\otimes V$$, there are scalars $$c_{ij}$$ such that \begin{align} \xi=\sum_{i=1}^n\sum_{j=1}^mc_{ij}u_i\otimes v_j \end{align}