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.
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?
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).
You should view AB as a collection of dot products ie. ab_{11} (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.