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Multiplication Of Vectors with a Vector As An Element

  1. Jun 29, 2012 #1
    Hi,

    If I have a vector c = [ a , 1 ]T where the element a is a vector. If I multiply c by its transpose:

    cTc

    is this defined? How do I calculate the a*a? Matrix multiplication rules would say that a * a is undefined because it is a nx1 matrix multiplied by a nx1 matrix. Or is the convention that it is the dot product?

    Thank you in advance for your help!
     
  2. jcsd
  3. Jun 29, 2012 #2

    HallsofIvy

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    In order to write vectors like that, you will need to think of the original vector, v, as a column matrix, say, [itex] v= \begin{bmatrix} x\\ y\end{bmatrix}[/itex] so that its transpose is a row matrix: [itex]v*= \begin{bmatrix}x & y\end{bmatrix}[/itex]. Then the product, v*u, is the matrix product [itex]\begin{bmatrix} x & y\end{bmatrix}\begin{bmatrix}a \\ b \end{bmatrix}= ax+ by[/itex].

    Of course, that is only notation. If you want to write the vector v as a row, fine. Then its transpose is a column and you have to write the product in the other order. Or you can go with the more abstract concept- given a vector in a vector space, V, there exist an isomorphism from V to its dual, V*, the set of linear functions from V to the underlying field. In that case "v*u" is the linear function corresponding to v applied the vector u.
     
  4. Jun 29, 2012 #3
    Are you sure this is a well-defined vector, though?
    The matrix notation, [itex]\vec{u} = \begin{bmatrix}x_1 \\ x_2 \\ \vdots \\ x_n\end{bmatrix}[/itex] is defined such that [itex]x_1, \dots, x_n[/itex] are the components of your vector in a given basis of your vector space, so they must be scalars.
    That's why I don't think having a vector there makes much sense.
     
  5. Jun 29, 2012 #4
    Thank you for the replies...so then does, a, have to be a column vector?
     
  6. Jun 29, 2012 #5
    when you take the dot product of two vectors a.b what you're really doing is the transpose of a and then doing matrix multiplication

    so [itex] a.b === a^t b[/itex]

    if a is 1xn and b is nx1 then [itex]a^t[/itex] is 1xn and so the matrix multiplication is defined (and gives you back a 1x1 matrix aka a real number)
     
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