What is the best way to remove empty rows and columns of an asymmetric square matrix if and only if they are both empty for a given index. That is, it should remove row AND column i if and only if both row i AND column i are all zeros. The result will be a square matrix. This is useful for reducing the size of directed adjacency matrices for graphs or Markov models among other things.

This question is similar to this question, but requires a solution that works when we can’t satisfy the condition: “when the i-th row is zeros, then i-th column is also zeros.” I believe lacking this constraint makes the DeleteCases and Select-based methods unusable as they are offered. But there may be a way to adapt them to this more general case.

Currently my best method is the following:

First find the indices for which the respective matrix row AND column are zeros.

m = SomeSquareMatrix; n = Length[m]; ids=Table[If[Total[m[[i]]]==0&&Total[m[[All,i]]]==0,{i},##&[]],{i,n}]; 

Note that Total works great for probabilities and edge weights that are always positive, but could be replaced by

ids=Table[If[m[[i]]==ConstantArray[0, n]&&Flatten[m[[All,i]]]==ConstantArray[0, n],{i},##&[]],{i,n}]; 

The ##&[] means that if the condition is false, return nothing. One you’ve identified the empty row-column combos, delete them:

m=Transpose[Delete[Transpose[Delete[m,ids]],ids]]; 

The last line is basically the same method used for this question and similar to the DeleteCases approach, so it might actually be the best way to go. However, I suspect there is a superior method that combines identifying and removing the empty pairs of rows and columns.