Sort matrix rows and columns while keeping heading

Question

I would like to sort columns and rows of a matrix in descending order by the sum of each row/column. However, my problem is that I cannot do so because I need to keep the heading of every column/row to track how they change when ordering the values.

A part of the matrix thereby looks as follows:

{ {"", "A", "B", "C", "D"}
, {"A", 3784.77, 14.619, 154191., 1563.92}
, {"B", 27.8608, 29.7455, 148986., 414.57}
, {"C", 38936.7, 2059.22, 1.06501*10^9, 33946.7}
, {"D", 18.3014, 428.637, 2.12249*10^7, 24287.4}
}

As explained, now I would like to sum the rows and columns up, keeping the labels (A to D) and then order them from the largest to the smallest, first ordering the rows, then the columns. Can someone please help me?

Thank you in advance!

Best regards Alex

Answer 1

You can separate headers and use SortBy with a custom ordering function:

sortRows[matr_] := Join[{First[matr]}, SortBy[Rest[matr], (-Total[Rest[#]]) &]];
sortCols[matr_] := Transpose@sortRows@Transpose[matr];

sortCols[sortRows[matrix]]

This will result in:

{{"", "C", "D", "A", "B"},
{"C", 1.06501*10^9, 33946.7, 38936.7, 2059.22},
{"D", 2.12249*10^7, 24287.4, 18.3014, 428.637},
{"A", 154191., 1563.92, 3784.77, 14.619},
{"B", 148986., 414.57, 27.8608, 29.7455}}

Answer 2

The reason to post this answer is that although it is longer than a dedicated answer, its code is more systematic (and bureaucratic) through the use of the package RSparseMatrix.

Import["https://raw.githubusercontent.com/antononcube/\
MathematicaForPrediction/master/Misc/RSparseMatrix.m"]

matTbl = {{"", "A", "B", "C", "D"}, {"A", 3784.77, 14.619, 154191., 
    1563.92}, {"B", 27.8608, 29.7455, 148986., 414.57}, {"C", 38936.7,
     2059.22, 1.06501*10^9, 33946.7}, {"D", 18.3014, 428.637, 
    2.12249*10^7, 24287.4}};

rmat = ToRSparseMatrix[SparseArray[matTbl[[2 ;; -1, 2 ;; -1]]], 
  "RowNames" -> matTbl[[2 ;; -1, 1]], 
  "ColumnNames" -> matTbl[[1, 2 ;; -1]]];

MatrixForm[rmat]

rord = Ordering[-RowSums[rmat]];
cord = Ordering[-ColumnSums[rmat]];
MatrixForm[rmat[[rord, cord]]]

matTbl = Normal[SparseArray[rmat1]];
matTbl = MapThread[Prepend, {matTbl, RowNames[rmat1]}];
matTbl = Prepend[matTbl, Prepend[ColumnNames[rmat1], ""]];
Grid[matTbl]

(Also, see my answer to a related question, How to merge Datasets representing matrices with named rows/columns?.)

Answer 3

You can directly operate on the sub matrix:

matrix[[2 ;;, 2 ;;]] = 
  SortBy[Transpose@SortBy[Transpose@matrix[[2 ;;, 2 ;;]], Total],Total];
MatrixForm[m]