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Tsampirlis Chapter 1 Inner Product

  1. Feb 16, 2016 #1
    Hi all,

    tyfAGOW.jpg

    The basis vectors are defined as 1x3 matrices, how can the result be a 3x3 matrix?

    How can the result of a dot product be a 3x3 matrix, I'm stumbled, how can I evaluate this?

    A inner product returns a scalar, and now it returns a 3x3 matrix, please help.

    Thanks.
     
  2. jcsd
  3. Feb 16, 2016 #2

    Samy_A

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    You have 9 dot products.
    You have a term ##g_{\mu\nu}## for all the possible combinations of ##\mu, \ \nu##. That's 3*3.

    The 3*3 matrix is :
    ##\begin{pmatrix}
    e_1\cdot e_1 & e_1\cdot e_2 & e_1\cdot e_3\\
    e_2\cdot e_1 & e_2\cdot e_2 & e_2\cdot e_3\\
    e_3\cdot e_1 & e_3\cdot e_2 & e_3\cdot e_3
    \end{pmatrix}##
     
  4. Feb 16, 2016 #3
    Apparently I have never heard of a matrix of an inner product.



    Should I follow this?
     
  5. Feb 16, 2016 #4
    But in the book they define lower indices as the number of colums, so I thought it were two 1x3 matrices?
     
  6. Feb 16, 2016 #5

    Samy_A

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    An inner product associates a pair of vectors with a scalar (and has a number of properties that I won't write down here).
    So with a basis of 3 vectors, you can associate 3*3 inner products, and that gives you the matrix I posted above.
    I don't know about the notation in your book, but in the image you posted it is clearly stated that ##e_1, e_2, e_3## form a basis, thus each of them is a vector. Their respective inner products is a perfectly well defined scalar.

    I only watch the beginning of the video, but yes, the matrix representation of a inner product she is computing is the same concept as the one in your book. She uses a somewhat different notation, though.
     
  7. Feb 16, 2016 #6
    So suddenly the lower indices represent columns in the case of μ and rows in the case for ν? And it's just plain matrix multiplication.
    First thing Samparlis did was define this in the book:

    nIX9L4j.jpg
     
  8. Feb 16, 2016 #7
    So when I saw eu and ev I thought two 1x3 matrices, since see notation above, which is not right, right?
     
  9. Feb 16, 2016 #8
    So my error is, I don't understand the notations? Lower indices are not always columns?


    If he defines the upper indices as rows and columns lower indices why isn't it


    guv = eu . ev

    instead of writing everything lower index
     
  10. Feb 16, 2016 #9

    Samy_A

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    Forget the notation for a moment. (I'm looking at the book right now, trying to understand it.)
    More important: do you understand how the combination of a basis (3 vectors) and an inner product leads to the definition of the matrix representation of that inner product?
     
  11. Feb 16, 2016 #10
    Apparently not, but in the video I posted it is defined as 2x2 matrices and I don't have an example for a 3x3 matrix.

    I have notion of Linear Algebra, never encountered matrix representation of an inner product.
     
  12. Feb 16, 2016 #11
  13. Feb 16, 2016 #12

    Samy_A

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    ##e_1, e_2, e_3## are (basis) vectors. They are not 1*3 matrices. The 1*3 matrix is ##(e_1, e_2, e_3)##, the matrix consisting of the three vectors taken together.
    There is nothing very difficult with this notion.
    If you get it in two dimensions, it is conceptually the same in 3 or 4 or 10 dimensions.

    He defines ##g_{\mu\nu}## as the inner product of the vectors ##e_\mu## and ##e_\nu##.
     
  14. Feb 16, 2016 #13
    So what he meant is

    e1
    e2 times e1 e2 e3
    e3

    and ordinary matrix multiplication?
     
  15. Feb 16, 2016 #14

    Samy_A

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    You can represent it this way. Just keep in mind that ##e_1,e_2,e_3## are themselves vectors, not scalars. And that the "product" of two vectors is the inner product.
     
  16. Feb 19, 2016 #15
    To come back on my question.

    Part of the confusion arose, because I've forgotton or looked over the meaning of the ≡ character. They were referring to the elements of the matrix. Stupid of me, but it's clear now.
     
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