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Dot Product 2x2 Matrix 
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#1
Apr2913, 10:27 AM

P: 40

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. 


#2
Apr2913, 11:05 AM

Sci Advisor
P: 839

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.



#3
Apr2913, 11:30 AM

Mentor
P: 15,206

\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} 


#4
Apr2913, 12:02 PM

P: 40

Dot Product 2x2 Matrix
Perfect thanks a lot!



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