How to Understand Rank of Tensor

Question

I want to understand rank of a Tensor by from a given matrix. Let assume we have a matrix

\begin{pmatrix} x_2^2 & x_1 x_2 \\ x_1 x_2 & x_1^2 \end{pmatrix}

Which we have generated from a cartesian components vector. I know a matrix representation from a vector would be rank one. But how to show this property from the matrix?

Answer 1

Let $u \in \mathbb{R}^m$ and $v \in \mathbb{R}^n$, then $uv^T$ is an $m\times n$ matrix called the outer product of $u$ and $v$. If $u$ and $v$ are both non-zero, $uv^T$ is not the zero matrix and has rank one. Conversely, any rank one $m\times n$ matrix can be written as an outer product. More generally, the rank of a matrix $A$ is the smallest number $k$ such that $A = u_1v_1^T + \dots + u_kv_k^T$ for some $u_1, \dots, u_k \in \mathbb{R}^m$, $v_1, \dots, v_k \in \mathbb{R}^n$.

Returning to your example, we have

$$\begin{bmatrix}x_2^2 & x_1x_2\\ x_1x_2 & x_1^2\end{bmatrix} = \begin{bmatrix}x_2\\ x_1\end{bmatrix}[x_2\ \ x_1]$$

so it has rank one.