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 tensot 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, p 30 (48 in pdf file) example 1,6. In this example is given a calculation of a rank of T and these decompostions without explanation.
Can someone help me to understand the examplle?
The formula on page 35 of your linked PDF (printed as page 17) gives $$ \begin{pmatrix} 1\\ 1 \end{pmatrix}\circ \begin{pmatrix} 1\\ 1 \end{pmatrix}\circ\begin{pmatrix} 1\\ 1 \end{pmatrix} = \begin{matrix}&&1&1\\&&1&1\\1&1&&\\1&1&&\end{matrix}\\ \begin{pmatrix} -1\\ 1 \end{pmatrix}\circ\begin{pmatrix} 1\\ -1 \end{pmatrix}\circ\begin{pmatrix} -1\\ 1\end{pmatrix} = \begin{matrix}&&-1&1\\&&1&-1\\1&-1&&\\-1&1&&\end{matrix} $$ Adding these together (element-wise) gives $$ \begin{matrix}&&0&2\\&&2&0\\2&0&&\\0&2&&\end{matrix} $$ As for the decomposition of $T$, page 43 (printed 25) tells you to start with the expression $$ T = \color{red}{\begin{pmatrix} 1\\ 1 \end{pmatrix}}\circ \color{blue}{\begin{pmatrix} 1\\ 1 \end{pmatrix}}\circ\color{green}{\begin{pmatrix} 1\\ 1 \end{pmatrix}}+\color{red}{\begin{pmatrix} -1\\ 1 \end{pmatrix}}\circ \color{blue}{\begin{pmatrix} 1\\ -1 \end{pmatrix}}\circ\color{green}{\begin{pmatrix} -1\\ 1 \end{pmatrix}} $$ Then the columns of $A$ are simply the red columns (the first column of each of the two products), the columns of $B$ are the blue columns (the middle columns in each product) and the columns of $C$ are the green columns (the last column in each product).