tensor: outer product, representation, decomposition

May 2019
1
0
london
It is given a tensor:

$T=\begin{pmatrix}
1\\ 1
\end{pmatrix}\circ \begin{pmatrix}
1\\ 1
\end{pmatrix}\circ\begin{pmatrix}
1\\ 1
\end{pmatrix}+\begin{pmatrix}
-1\\ 1
\end{pmatrix}\circ\begin{pmatrix}
1\\ -1
\end{pmatrix}\circ\begin{pmatrix}
-1\\ 1
\end{pmatrix}$


1) Why is it possible to write the tensor T as:
$T=\begin{pmatrix}
2 &0 \\
0& 2
\end{pmatrix} and \begin{pmatrix}
0 &2 \\
2& 0
\end{pmatrix}$

it is given in example that I can represent the tensor T as a sum of the outer product of vector triples and as 2 matrices. I have computed the outer product of the vector triples, but I can't get the same result.
Can someone provide me detailed calculation?

2) T=[[ABC]]

$A=\begin{pmatrix}
1 &-1 \\
1& 1
\end{pmatrix}$

$B=\begin{pmatrix}
1 &1 \\
1& -1
\end{pmatrix}$

$C=\begin{pmatrix}
1 &-1 \\
1& 1
\end{pmatrix}$


**How to compute A, B, C?**

Later on the p 35 (53), example 2. or on p 36(54) 2.2.1 the vectors a,b,c are given without an explanation of how he/she competed them. In §2.2.1 it is given that "we set...." and it is all. No explanation of how they find them.


I have found examples in [Analysis of 2 × 2 × 2 Tensors][1], p 30 (48 in pdf file) example 1,6. In this example is given a calculation of a rank of T and these decompositions without explanation.

Can someone help me to understand the example?


[1]: http://e-spacio.uned.es/fez/eserv.php?pid=bibliuned:mastermatavanz10&dsID=Documento.pdf
 
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