# When is $x \otimes y = y \otimes x$? [duplicate]

Let $$X$$ be a vector space and $$x \otimes y \in X \otimes X$$. Under which conditions is $$x \otimes y = y \otimes x$$? Does it nessecarily follow that $$x = \lambda y$$ for some $$\lambda$$ in the underlying field?

• Given that $X$ is a finite-dimensional $k$-vector space, the tensor product $x \otimes y$ of the vectors $x, y$ in $X$ is the matrix whose $j$th column is the column vector $(x_i y_j),$ where $x_i$ and $y_j$ are the coefficients of $x$ and $y$ with respect to some basis. – Carlo May 19 at 16:26
• For a simple example, consider the $\mathbb R$-vector space $\mathbb R^2$ with vectors $x = (1, 2)$ and $y = (1, 1).$ We have that $x \otimes y = \begin{pmatrix} 1 & 1 \\ 2 & 1 \end{pmatrix}.$ – Carlo May 19 at 16:28
• @EricWofsey Yes! – Jannik Pitt May 19 at 16:51

You're not entirely correct (consider $$y=0, x\neq0$$). But yes, $$x\otimes y=y\otimes x$$ means that $$x$$ and $$y$$ are collinear / linearly dependent.
• Chose some suitable linear forms $\alpha$ and $\beta$ and evaluate $\alpha\otimes \beta$ on $x\otimes y$ – DIdier_ May 19 at 16:27