Conditional sorting of columns in a matrix

Question

I have a matrix with ten columns, and I want to rearrange these columns with three conditions.

1. Column 1+2n and 2+2n (0<=n=<4) always remain together and in the same order (1+2n before 2+2n). Now we can say that we have five sets of columns, each having two columns.

2. The 2+2n th columns should be sorted with respect to the first elements in them.

3. In case first elements of any two 2+2n columns are same, they should be arranged with respect to the first elements of the corresponding 1+2n columns.

For example if the simplified matrix has only one row {1, 4, 7, 5, 8, 5, 0, 2, 3, 9}, it should be rearranged to {0, 2, 1, 4, 7, 5, 8, 5, 3, 9}.

Thanks for the help!

Answer 1

Update: A generalization that takes partition sizes, reference row and column sort order for sorting:

ClearAll[partitionedOrderingBy]
partitionedOrderingBy[m_, referenceRow_: 1, partitionSize_: 2, sortOrder_: Automatic] :=
 Module[{
  partitionedIndices = Partition[Range@Dimensions[m][[2]], partitionSize],
   sortFunctions = Table[With[{i = i}, m[[referenceRow, #[[i]]]] &], 
     {i, sortOrder /. Automatic -> Reverse @ Range @ partitionSize}]},
  Flatten @ SortBy[sortFunctions] @ partitionedIndices]

Examples:

SeedRandom[1]
mat = RandomInteger[10, {3, 12}];

MatrixForm @ mat

mat[[All, partitionedOrderingBy[mat]]]

highlightPartitions[m_, partitionSize_: 2] := MapIndexed[
 Highlighted[Style[#, 16], Background->ColorData[97]@(Ceiling[#2[[2]]/partitionSize])]&,
 m, {2}]

MatrixForm[#, TableHeadings -> {{1, 2, 3}, Range@12}] & @ highlightPartitions[mat]

Sort using default values for arguments:

With[{r = 1}, 
 MatrixForm[highlightPartitions[mat, 2][[All, #]], 
    TableHeadings -> {Range[3] /. r -> Style[r, 20, Bold, Red], #}] &@
  partitionedOrderingBy[mat]]

Use the second row as reference row for sorting:

With[{r = 2}, 
 MatrixForm[highlightPartitions[mat, 2][[All, #]], 
    TableHeadings -> {Range[3] /. r -> Style[r, 20, Bold, Red], #}] &@
  partitionedOrderingBy[mat, r]]

Use the lexical order {1,2} instead of the default {2,1}:

With[{r = 1}, 
 MatrixForm[highlightPartitions[mat, 2][[All, #]], 
    TableHeadings -> {Range[3] /. r -> Style[r, 20, Bold, Red], #}] &@
  partitionedOrderingBy[mat, r, 2, {1, 2}]]

Partition columns into consecutive triples (instead of consecutive pairs):

MatrixForm[#, TableHeadings -> {{1, 2, 3}, Range@12}] & @ highlightPartitions[mat, 3]

Use the default reference row (1) for sorting:

With[{r = 1}, 
 MatrixForm[highlightPartitions[mat, 3][[All, #]], 
    TableHeadings -> {Range[3] /. r -> Style[r, 20, Bold, Red], #}] &@
  partitionedOrderingBy[mat, r, 3]]

Use row 3 as the reference row:

With[{r = 3}, 
 MatrixForm[highlightPartitions[mat, 3][[All, #]], 
    TableHeadings -> {Range[3] /. r -> Style[r, 20, Bold, Red], #}] &@
  partitionedOrderingBy[mat, r, 3]]

MatrixForm @ highlightPartitions[mat, 3][[All, partitionedOrderingBy[mat, 2, 3]]]

Use partition size 4 and first row as reference row:

MatrixForm[#, TableHeadings -> {{1, 2, 3}, Range@12}] &@ highlightPartitions[mat, 4]

With[{r = 1}, 
 MatrixForm[highlightPartitions[mat, 4][[All, #]], 
    TableHeadings -> {Range[3] /. r -> Style[r, 20, Bold, Red], #}] &@
  partitionedOrderingBy[mat, r, 4]]

Original answer:

For input matrix m, Partition the column indices and lexically sort the pairs of indices based on first row elements of m )so that the index pair {pi1, pi2} comes before the pair {pj1, pj2} iff m[[1,pi2]] < m[[1,pj2]] or m[[1,pi2]] == m[[1,pj2]] and m[[1,pi1]] < m[[1,pj1]]):

ClearAll[partitionAndSort]

partitionAndSort[m_] := Module[{
   partitionedColumnIndices = Partition[Range @ Dimensions[m][[2]], 2], 
   lexicalSortFunctions = {m[[1, #[[2]]]] &, m[[1, #[[1]]]] &},
   columnsOrdering},
  columnsOrdering =  Flatten @ SortBy[lexicalSortFunctions] @ partitionedColumnIndices;
  m[[All, columnsOrdering]]]

Example:

SeedRandom[1]
mat = RandomInteger[10, {3, 10}];

MatrixForm @ mat

MatrixForm @ partitionAndSort @ mat