Exercise 1.12.10

Let $\gamma =(1,1),\alpha =(1,1),\beta =(1,1)$ and $r=(-1,0),p=(1,0),q=(1,0)$, we have

$T_1=\gamma \times \alpha \times \beta =\gamma \times \left[\begin{array}{cc}\hfill 1\hfill & \hfill 1\hfill \\ \hfill 1\hfill & \hfill 1\hfill \end{array}\right]=\left[\begin{array}{ccccc}\hfill 1\hfill & \hfill 1\hfill & \hfill \text{|}\hfill & \hfill 1\hfill & \hfill 1\hfill \\ \hfill 1\hfill & \hfill 1\hfill & \hfill \text{|}\hfill & \hfill 1\hfill & \hfill 1\hfill \end{array}\right]$

$T_2=r\times p\times q=r\times \left[\begin{array}{cc}\hfill 1\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill 0\hfill \end{array}\right]=\left[\begin{array}{ccccc}\hfill -1\hfill & \hfill 0\hfill & \hfill \text{|}\hfill & \hfill 0\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill 0\hfill & \hfill \text{|}\hfill & \hfill 0\hfill & \hfill 0\hfill \end{array}\right]$

So $T=T_1+T_2=\left[\begin{array}{ccccc}\hfill 0\hfill & \hfill 1\hfill & \hfill \text{|}\hfill & \hfill 1\hfill & \hfill 1\hfill \\ \hfill 1\hfill & \hfill 1\hfill & \hfill \text{|}\hfill & \hfill 1\hfill & \hfill 1\hfill \end{array}\right]$

Note, to find such vectors, you can set the components of these vectors to variables, and solve for them using the condition that $T$ only has 1 zero entry.

The closest rank-1 tensor to $T$ is the $T_1$.