Matrix multiplication vs dot product

1. jabers

15
What is the difference between matrix multiplication and the dot product of two matrices? Is there a difference?

If,

$$A = \begin{pmatrix} a & b \\ c & d \end{pmatrix}$$

and

$$B = \begin{pmatrix} e & f \\ g & h \end{pmatrix}$$

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

and

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

? Is this correct? Any help would be appreciated.

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3. sharks

836
Don't confuse dot product of matrix with vectors. The second product is correct.

4. jabers

15
so,

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

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

5. deluks917

383
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).

6. JoshMaths

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.