matrix multiplication vs dot product


by jabers
Tags: dot product, matrices
jabers
jabers is offline
#1
May21-12, 09:25 AM
P: 14
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.
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sharks
sharks is offline
#2
May21-12, 10:19 AM
PF Gold
sharks's Avatar
P: 836
Don't confuse dot product of matrix with vectors. The second product is correct.
jabers
jabers is offline
#3
May21-12, 10:24 AM
P: 14
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?

deluks917
deluks917 is offline
#4
May21-12, 11:35 AM
P: 367

matrix multiplication vs dot product


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).
JoshMaths
JoshMaths is offline
#5
May21-12, 01:33 PM
P: 31
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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