Dot Product 2x2 Matrix

  1. 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. jcsd
  3. pwsnafu

    pwsnafu 902
    Science Advisor

    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.
     
  4. D H

    Staff: Mentor

    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}
     
  5. Perfect thanks a lot!
     
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