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!

 

[CellCoree]

The dot product of two 2x2 matrices is a mathematical operation that results in a scalar value, not a matrix. It is also known as the inner product or scalar product. The formula for the dot product of two matrices is the sum of the products of their corresponding elements. In other words, the dot product of two matrices A and B is given by:

\begin{equation}
A \cdot B = \sum_{i=1}^{n} \sum_{j=1}^{m} a_{ij}b_{ij}
\end{equation}

In your example, the dot product of the two 2x2 matrices would be:

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

It is important to note that the dot product of two matrices is only defined when the number of columns in the first matrix is equal to the number of rows in the second matrix. This is why your teacher mentioned that the basis images (matrices) are orthonormal, meaning they are perpendicular to each other and have a length of 1. In this case, the dot product of any two basis images (matrices) would always be 0, as the sum of their corresponding elements would always be 0. This is a useful property in image processing as it allows for efficient and accurate calculations. I hope this helps clarify the concept of the dot product of two 2x2 matrices.