What is the Dot Product of Two 2x2 Matrices?

[Owen-]

This seems like a very basic question that I should know the answer to, but in my image processing class, my teacher explained that a basis set of images(matrices) are orthonormal.

He said that the DOT product between two basis images (in this case two 2x2 matrices) is 0. so, for example

\begin{equation}
\begin{bmatrix}
a & b\\
c & d
\end{bmatrix}
\cdot
\begin{bmatrix}
e & f\\
g & h
\end{bmatrix}
=0
\end{equation}

I don't understand how this can be. I always thought it gave another matrix, and not a direct value:
\begin{equation}
\begin{bmatrix}
a & b\\
c & d
\end{bmatrix}
\cdot
\begin{bmatrix}
e & f\\
g & h
\end{bmatrix}
=
\begin{bmatrix}
ae+bg & af+bh\\
ce+dg & cf+dh
\end{bmatrix}
\end{equation}

Can someone help me out? It would be unbelieveably helpful,
Thanks!
Owen.

 

[pwsnafu]

The only possibility I can think of is to take a 2x2 matrix and write it out in the form $a e_{11} + b e_{12} + c e_{21} + d e_{22}$, ie as a four dimensional vector space. Then the e's form an orthonormal basis.

 

[D H]

I don't understand how this can be. I always thought it gave another matrix, and not a direct value:
\begin{equation}
\begin{bmatrix}
a & b\\
c & d
\end{bmatrix}
\cdot
\begin{bmatrix}
e & f\\
g & h
\end{bmatrix}
=
\begin{bmatrix}
ae+bg & af+bh\\
ce+dg & cf+dh
\end{bmatrix}
\end{equation}

Can someone help me out? It would be unbelieveably helpful,
Thanks!
Owen.

That's the matrix product, not the dot product. A dot product (inner product) is a scalar. Always. For matrices, the typical definition of the dot product is the Frobenius inner product. Simply compute as if the matrix was a vector. For real matrices,

\begin{equation}
A\cdot B \equiv \sum_i \sum_j A_{ij} B_{ij}
\end{equation}
For your pair of 2x2 matrices,
\begin{equation}
\begin{bmatrix}
a & b\\
c & d
\end{bmatrix}
\cdot
\begin{bmatrix}
e & f\\
g & h
\end{bmatrix}
= ae + bf + cg + dh\end{equation}

 

[Owen-]

Perfect thanks a lot!