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Matrix Multiplication & Norms

  1. Oct 3, 2012 #1
    1. Why is the norm of a vector noted by double pipes when it is just the magnitude which is notated by single pipes?

    2. Does anyone know where I could find out why matrix multiplication is defined the way it is? I know how to do it, but I do not understand why it is that way.

    Thank you.
     
  2. jcsd
  3. Oct 3, 2012 #2
    1) You define the norm of a vector as [itex]||\vec{v}||[/itex] instead of just [itex]|\vec{v}|[/itex] because, although they often coincide, this is not necessary true. In some spaces (different from euclidean ones) you can define different kinds of norm, as for example: [itex]||\vec{v}||=\min |\vec{v(x)}| [/itex], where x is a certain variable.

    2) The way the matrix product works is a definition. You define an operation and then you find al the consequences.
     
  4. Oct 3, 2012 #3

    Fredrik

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    1. In this case, it doesn't matter if you write ##\|x\|## or ##|x|##. The ##\|\,\|## notation is standard for norms, and the map ##x\mapsto|x|## is a norm. This is why it makes sense to write ##\|x\|## instead of ##|x|##.
    2. For this you need to understand the relationship between linear operators and matrices. See e.g. this post. (You can ignore the quote and the stuff below it). Make sure that you understand bases of vector spaces before you try to understand this. The motivation behind the definition of matrix multiplication is what I said in this post. I think I have posted more details of this somewhere else. I will try to find it... Edit: There it is.

    Edit 2: Some additional comments:
    • Linear operator, linear transformation, linear map, linear function all mean the same thing. (Most of the time anyway. Some authors use the term "operator" only when the domain and codomain are the same vector space, and some use the term "function" only when the codomain is ##\mathbb R## or ##\mathbb C##).
    • The point of view that I'm advocating in these posts is that for each linear transformation ##A:U\to V##, and each pair of bases (one for U, one for V), there's a matrix [A] that corresponds to it in the following way: We take the number on row i, column j to be ##(Au_j)_i##. Here ##u_j## is the jth member of the given basis for U, and ##(Au_j)_i## is the ith component of the vector Au_j, in the given basis for V. Matrix multiplication is defined the way it is to ensure that we always have ##[A\circ B]=[A]##.
     
    Last edited: Oct 3, 2012
  5. Oct 3, 2012 #4
    Thank you.

    I understand it is a definition. But I'm asking why it is defined that way.

    EDIT:
    Thank you Fredrik. I shall read through those posts, though I fear the mathematics is beyond me.
     
    Last edited: Oct 3, 2012
  6. Oct 3, 2012 #5

    Fredrik

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    The notation (in particular all the indices) may be intimidating, but if you understand vector spaces, bases, and what it means for a function to be linear, it's actually pretty easy. You need to know that the definition of matrix multiplication can be written as ##(AB)_{ij}=\sum_k A_{ik}B_{kj}##. The right-hand side is often written as ##A_{ik}B_{kj}##, especially by physicists, because it's easy enough to remember that there's always a sum over each index that appears twice.
     
  7. Oct 3, 2012 #6

    AlephZero

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    The basic reason is "because it is a very useful way to combine two matrices". A simple example is using matrices to represent simultanous equations. If you have equations like $$\begin{align} ax + by &= p\\ cx + dy &= q\end{align}$$ that corresponds to the matrix equation $$\begin{bmatrix} a & b \\ c & d\end{bmatrix} \begin{bmatrix} x \\ y \end{bmatrix} = \begin{bmatrix} p \\ q \end{bmatrix}$$.
    An apparently simpler definition of "multiplication" like$$\begin{bmatrix} a & b \\ c & d\end{bmatrix} \begin{bmatrix} p & q \\ r & s \end{bmatrix} = \begin{bmatrix} ap & bq \\ cr & ds\end{bmatrix} $$ (similar to the definition of matrix addition) turns out to be much less useful.
     
  8. Oct 3, 2012 #7

    HallsofIvy

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    C'mon, don't delete those, it was the "best explanation ever"!
     
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