What is the geometric interpretation of this matrix?

Question

Let $x$ be some vector in two dimensions. To make my question clearer, let me start with an example where I know the solution: The matrix \begin{equation} \frac{1}{x_1x_1+x_2x_2}\begin{pmatrix}x_2x_2&-x_1x_2\\-x_2x_1&x_1x_1\end{pmatrix} \end{equation} is interpreted as the projection onto the subspace spanned by $x$ (more precisely, the representation w.r.t. some basis). That being said, what is the interpretation of the following matrix: \begin{equation} T:=\begin{pmatrix}x_2x_2&-x_1x_2\\-x_2x_1&x_1x_1\end{pmatrix} \end{equation} For example, we may think of $T$ as the representation of a vector space endomorphism (as above) or a bilinear form w.r.t. a basis. The matrix $T$ appears in the discussion of tensors in Riley, Hobson and Bence (3rd).

Answer 1

Yes, as it can be written under the following matrix form:

$$\binom{ \ \ x_1}{-x_2}(x_1 \ \ -x_2) $$

which, being the product of column vector and a row vector is interpretable as a tensorial expression from $$(\mathbb{R}^2) \otimes (\mathbb{R}^2)^*$$

Edit 1: Why row vectors can be identified with linear mappings $(x,y) \in \mathbb{R}^2 \to ax+by \in \mathbb{R}$ ? Because one can write them :

$$\binom{x}{y} \ \to \ (a \ \ b)\binom{x}{y}$$

Edit 2: In fact, there is a second reason for assimilating the given matrix to a tensor product : it is the fact that it can be written

$$(x_1 \ \ -x_2) \oplus \binom{x_1}{-x_2}$$

where this time $\oplus$ denotes the Kronecker product of 2 matrices.