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A question about tensor product

  1. Oct 8, 2015 #1
    Hey it might be a stupid question but I saw that the tensor product of 2 vectors with dim m and n gives another vector with dimension mn and in another context I saw that the tensor product of vector gives a metrix. For example from sean carroll's book: "If T is a (k,l) tensor and S is a (m, n) tensor, we define a (k + m, l + n) tensor T ⊗ S"
    so the tensor product of two type 1 tensors,k=1,vectors, is a metrix
    and in the context of quantum mechanic I saw
    (1,0)⊗(1,0)↦(1,0,0,0) when those our basis vectors.
    I'm sure I'm just getting something wrong but I am hopefull that you can explain me what.
     
  2. jcsd
  3. Oct 8, 2015 #2
    Why it would be? If one of them is dual vector, then it might be. I will give you some examples:
    [tex] \begin{pmatrix} 1 \\ 2 \end{pmatrix} \otimes \begin{pmatrix} 1 & 2 \end{pmatrix} = \begin{pmatrix} 1 \times \begin{pmatrix} 1 & 2 \end{pmatrix}\\ 2 \times \begin{pmatrix} 1 & 2 \end{pmatrix} \end{pmatrix} = \begin{pmatrix} 1 & 2 \\ 2 & 4 \end{pmatrix}, [/tex]
    [tex] \begin{pmatrix} 1 & 2 \end{pmatrix} \otimes \begin{pmatrix} 1 & 2 \end{pmatrix} = \begin{pmatrix} 1 \times \begin{pmatrix} 1 & 2 \end{pmatrix} & 2 \times \begin{pmatrix} 1 & 2 \end{pmatrix} \end{pmatrix} = \begin{pmatrix} 1 & 2 & 2 & 4 \end{pmatrix} [/tex]
     
  4. Oct 8, 2015 #3
    Because by the defenition of sean, let's take a type (1,0) tensor which is a vector and another (1,0) tensor which is also a vector and the product will be a (2,0) tensor, which is a metrix.
     
    Last edited: Oct 8, 2015
  5. Oct 8, 2015 #4

    WWGD

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    A vector can be seen as a ## 1 \times n ## matrix.
     
  6. Oct 8, 2015 #5
    That is true. But, the results of mine are matrices with the size [itex] 2 \times 2 [/itex] and [itex] 1 \times 4 [/itex].
     
  7. Oct 8, 2015 #6

    WWGD

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    Actually, if your map is k-linear ( in any " coordinate") for k>2 (where you may have quadratic forms), it is not representable as a matrix anymore. That is the actual point of tensors: to represent k - , or j- ( k,j pos. integers) linear maps in many variables, which is not feasible with matrices alone whenever you have an index >2.
    Only linear and bilinear maps may be represented using matrices.
     
  8. Oct 8, 2015 #7

    HallsofIvy

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    I feel compelled to point out that a tensor can be represented by a matrix in a given coordinate system, but, strictly speaking, a tensor is NOT a matrix.
     
  9. Oct 8, 2015 #8

    WWGD

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    How do you represent a higher-order tensor as a matrix? e.g., a 3-linear map .
     
  10. Oct 15, 2015 #9

    HallsofIvy

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    In the same sense that a "vector" is a 1 by 3 matrix, so a higher order tensor can be represented by a "3 by 3 by 3" or higher matrix. I admit that is stretching the concept of "matrix" a bit far. My point was simply that a matrix is NOT a tensor.
     
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