tensor: outer product, representation, decomposition

nanito489

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 Â§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 Ã— 2 Ã— 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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