Inverse matrix notation question

[SamanthaYellow]

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

Personally I interpret $(M_{ij})^{-1}$ in the usual sense of an inverse matrix, where it would have the property $\textbf M \textbf M^{-1} = I$, but perhaps there are other interpretations that I don't know about. Thanks!

 

[disregardthat]

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.

 

[Fredrik]

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.

 

[SamanthaYellow]

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!

 

[blue_raver22]

I can confirm that your interpretation of (M_{ij})^{-1} as the inverse matrix is correct. The notation (M_{ij})^{-1} is commonly used to represent the inverse of a matrix, defined as the matrix that when multiplied by the original matrix, results in the identity matrix. This notation is widely accepted and understood in the scientific community.

While it is possible to interpret (M_{ij})^{-1} as a matrix with elements 1/m_{ij}, this would not be the standard or conventional interpretation. In fact, using this notation could lead to confusion and misinterpretation, as it goes against the established meaning of inverse matrices.

It is important to use standard notation and definitions in mathematical and scientific discussions to avoid misunderstandings and ensure clear communication. In this case, (M_{ij})^{-1} should be interpreted as the inverse matrix, unless otherwise specified. I hope this settles your argument and clarifies the correct interpretation of this notation.