My Math Forum tensor: outer product, representation, decomposition

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 May 30th, 2019, 11:39 PM #1 Newbie   Joined: May 2019 From: london Posts: 1 Thanks: 0 tensor: outer product, representation, decomposition 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.ph...=Documento.pdf Last edited by skipjack; May 31st, 2019 at 03:18 AM.

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