Matrix multiplication vs dot product

[jabers]

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

If,

$$A =<br /> \begin{pmatrix}<br /> a & b \\<br /> c & d<br /> \end{pmatrix}$$

and

$$B =<br /> \begin{pmatrix}<br /> e & f \\<br /> g & h<br /> \end{pmatrix}$$

then does
$${\mathbf{A} \cdot \mathbf{B}} =<br /> \begin{pmatrix}<br /> ae & bf \\<br /> cg & dh<br /> \end{pmatrix}$$

and

$$AB = <br /> \begin{pmatrix}<br /> ae + bg & af + bh \\<br /> ce + dg & cf + dh<br /> \end{pmatrix}$$

? Is this correct? Any help would be appreciated.

 

[DryRun]

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

 

[jabers]

so,

$${\mathbf{A} \cdot \mathbf{B}} = AB = <br /> \begin{pmatrix}<br /> ae + bg & af + bh \\<br /> ce + dg & cf + dh<br /> \end{pmatrix}$$

With matrices the dot product means that you need to multiply the matrices? Correct?

 

[deluks917]

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$\bullet$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]

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.