Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

Inverse matrix notation question

  1. Apr 17, 2014 #1
    I'm hoping that you can help me settle an argument. For a matrix [itex]\textbf{M}[/itex] with elements [itex]m_{ij}[/itex], is there any sitaution where the notation [itex](M_{ij})^{-1}[/itex] could be correctly interpreted as a matrix with elements [itex]1/m_{ij}[/itex]?

    Personally I interpret [itex](M_{ij})^{-1}[/itex] in the usual sense of an inverse matrix, where it would have the property [itex]\textbf M \textbf M^{-1} = I[/itex], but perhaps there are other interpretations that I don't know about. Thanks!
  2. jcsd
  3. Apr 17, 2014 #2


    User Avatar
    Science Advisor

    The only interpretation I can think of is when you define matrix multiplication component-wise. In this case, the invertible matrices would be the ones with non-zero values, and the matrix you describe would be the inverse. Note that the identity matrix would be the one with 1's as values. I don't think this is commonly used.
  4. Apr 17, 2014 #3


    User Avatar
    Staff Emeritus
    Science Advisor
    Gold Member

    I'm not a fan of the notation ##(M_{ij})^{-1}##, mainly because ##M_{ij}## should refer to the component on row i, column j, not the matrix itself. I'm also not a fan of using a lowercase m for the components, because that prevents us from writing the definition of matrix multiplication in what I consider the obviously best way: ##(AB)_{ij}=\sum_k A_{ik}B_{kj}##. I find it very puzzling that some authors go out of their way to avoid this notation, by writing things like "if ##C=AB##, then ##c_{ij}=\sum_k a_{ik}b_{kj}##".

    If M is a diagonal matrix, for example
    \begin{pmatrix}2 & 0 & 0\\ 0 & 3 & 0\\ 0 & 0 & 4\end{pmatrix} then its inverse is simply
    \begin{pmatrix}\frac 1 2 & 0 & 0\\ 0 & \frac 1 3 & 0\\ 0 & 0 & \frac 1 4\end{pmatrix} But even if M is diagonal, and we use horrible notation, we still don't quite have ##(M_{ij})^{-1}=1/m_{ij}## because the off-diagonal elements of ##M^{-1}## aren't 1/0.
    Last edited: Apr 17, 2014
  5. Apr 18, 2014 #4
    This is helpful. The matrix in question isn't diagonal, and that's a good point about 1/0. Hopefully I can convince this other person to change their notation!
Share this great discussion with others via Reddit, Google+, Twitter, or Facebook