Matrix multiplication vs dot product

  1. What is the difference between matrix multiplication and the dot product of two matrices? Is there a difference?

    If,

    [tex]A =
    \begin{pmatrix}
    a & b \\
    c & d
    \end{pmatrix}[/tex]

    and

    [tex]B =
    \begin{pmatrix}
    e & f \\
    g & h
    \end{pmatrix}[/tex]

    then does
    [tex]
    {\mathbf{A} \cdot \mathbf{B}} =
    \begin{pmatrix}
    ae & bf \\
    cg & dh
    \end{pmatrix}[/tex]

    and

    [tex]AB =
    \begin{pmatrix}
    ae + bg & af + bh \\
    ce + dg & cf + dh
    \end{pmatrix}[/tex]

    ? Is this correct? Any help would be appreciated.
     
  2. jcsd
  3. sharks

    sharks 836
    Gold Member

    Don't confuse dot product of matrix with vectors. The second product is correct.
     
  4. so,

    [tex]{\mathbf{A} \cdot \mathbf{B}} = AB =
    \begin{pmatrix}
    ae + bg & af + bh \\
    ce + dg & cf + dh
    \end{pmatrix}[/tex]

    With matrices the dot product means that you need to multiply the matrices? Correct?
     
  5. Usually the "dot product" of two matrices is not defined. I think a "dot product" should output a real (or complex) number. So one definition of A[itex]\bullet[/itex]B is ae + bf + cg + df. This is thinking of A, B as elements of R^4. If we want our dot product to be a bi-linear map into R this is how we need to define it (up to multiplication by a constant).
     
  6. You should view AB as a collection of dot products ie.
    ab11 (top left of AB) can be described as the dot product of

    \begin{pmatrix}
    a & b
    \end{pmatrix}dot\begin{pmatrix}
    e \\
    g
    \end{pmatrix}

    and so on for the rest of the positions.
     
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