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Ordinary matrix-vector multiplication

  1. Mar 5, 2006 #1
    Let [tex]P=(p_{ij})[/tex] be a real symmetric 2x2 matrix. Show that the function on [tex]\mathbb{R}^2\times\mathbb{R}^2[/tex] (Where R^2 is a space of column vectors) defined by [tex]<v,w>=v^tPw[/tex] is an inner product if and only if [tex]p_{11}[/tex] and [tex]det(P)[/tex] are both swtrictly positive.

    I just need to know what [tex]Pw[/tex] means in [tex]<v,w>=v^tPw[/tex].
     
  2. jcsd
  3. Mar 5, 2006 #2

    arildno

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    Ordinary matrix-vector multiplication.
     
  4. Mar 5, 2006 #3
    I see. I thought it was "P of w".
     
  5. Mar 5, 2006 #4
    This function doesn't send vectors to scalars, it can't be an inner product, unless I understood something wrong.
     
  6. Mar 5, 2006 #5
    v and w are 1x2 column vectors, right? So why is this function on R^2xR^2? Isn't it defined on the vector space R^2?
     
  7. Mar 5, 2006 #6

    shmoe

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    they're 2x1, but there are two of them. This function takes a pair of vectors, (v,w) and gives a real number, so the domain is R^2xR^2.
     
  8. Mar 5, 2006 #7

    HallsofIvy

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    v is in R2, w is in R2 so (v, w) is in R2 x R2.
     
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