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Inner product vs. dot product

  1. May 18, 2006 #1
    Is there any difference between an inner product and a dot product?
     
  2. jcsd
  3. May 18, 2006 #2
  4. May 18, 2006 #3

    arildno

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    Yes:
    The dot product is an inner product, whereas "inner product" is the more general term.

    EDIT:
    I'm getting old. But then again, how could I ever compete with young, strong dragons swooping down on its prey? :frown:
     
  5. May 18, 2006 #4
    What would be an example of an inner product that it not a dot product?
     
  6. May 18, 2006 #5

    arildno

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    Well, as I'm used to it the term "dot product" is usually reserved for an operation on a finite Euclidean space; I haven't seen the term "dot product" being used for inner products defined on function spaces, for example.
     
  7. May 18, 2006 #6

    LeonhardEuler

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    Just expanding on what arildno said, one specific example is the Frobenius inner product of two matrices. It is defined by:
    <A,B>=tr(B*A)
    Where B* is the conjugate transpose, or adjoint of B.
     
  8. May 18, 2006 #7

    TD

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    For continous functions from [a,b] to R, you can define an inner product <.,.> as:

    [tex]\left\langle {f,g} \right\rangle = \int\limits_a^b {f\left( x \right)g\left( x \right)dx} [/tex]

    This is also an example on that Mathworld page.
     
  9. May 18, 2006 #8

    Hurkyl

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    (a, b) . (c, d) = ac + 2bd
    (a, b) . (c, d) = (a+b)(c+d) + (a-b)(c-d)

    (a) . (b) = 2ab
     
  10. May 19, 2006 #9
    Are these postulates?
     
  11. May 19, 2006 #10

    arildno

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    No, you can show that they satisfy the PROPERTIES of the inner product:
    To take the first:
    1. (a,b).(a,b)=a^2+2b^2>0 unless a=b=0
    2. (a,b).(c,d)=ac+2bd=ca+2db=(c,d).(a,b)
    and so on with the rest of an inner product's properties.
     
  12. May 19, 2006 #11

    Astronuc

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    I wish someone had explained that difference to me 30+ years ago, when I was first learning about vectors and multidimensional analysis. I got to university thinking that 'dot' and 'inner' product were the same, and it was terribly confusing when an inner product was introduced with a different meaning than a dot product.
     
  13. May 19, 2006 #12

    arildno

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    The way I sort of organized the concepts to myself was like this:
    General terms:
    Inner/Outer products
    Special terms:
    Dot/cross products
    Scalar/vector products

    This was at least helpful for me.
     
  14. May 19, 2006 #13
    There's an outer product now?
     
  15. May 19, 2006 #14
    Yup. It's commonly called the tensor or direct product.
     
  16. May 19, 2006 #15

    Hurkyl

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    No -- they are definitions of three different inner products. The first two are on R², and the third is on R.
     
  17. May 19, 2006 #16

    HallsofIvy

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    The theoretical "meat" of the Gram-Schmidt orthogonaliztion process is that any inner product is a dot product in some basis. Given an inner product, choose a basis and use Gram-Schmidt to derive an orthonormal basis {e1, e2,...,en}. For any vectors u,v, write u= a1e1+ ... , v= b1e1+... . Then the inner product <u,v>= a1b1+... + anbn.

    Take a look at Hurkyl's examples:
    (a, b) . (c, d) = ac + 2bd
    (a, b) . (c, d) = (a+b)(c+d) + (a-b)(c-d)

    (a) . (b) = 2ab
    Starting from the basis {(1, 0), (0,1)}, (or just (1) for the third example) use Gram-Schmidt with each of these inner products to derive an orthonormal basis and show that the inner product is the dot product in that basis.
     
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