Linear Algebra

Section I. Definition 3154.3 Lemma A row-swap in a permutation matrix changes the number of inversionsfrom even to odd, or from odd to even.Proof Consider a swap of rows j and k, where k > j. If the two rows areadjacent⎛ ⎞ ⎛ ⎞P φ =⎜⎝.ι φ(j)ι φ(k).⎟⎠ρ k ↔ρ j−→⎜⎝.ι φ(k)ι φ(j)then since inversions involving rows not in this pair are not affected, the swapchanges the total number of inversions by one, either removing or producing oneinversion depending on whether φ(j) > φ(k) or not. Consequently, the totalnumber of inversions changes from odd to even or from even to odd.If the rows are not adjacent then we can swap them via a sequence of adjacentswaps, first bringing row k up⎛ ⎞⎛ ⎞..ι φ(j)ι φ(k)⎜ι φ(j+1) ⎟⎜ ι φ(j) ⎟⎜⎝ι φ(j+2).ι φ(k).⎟⎠ρ k ↔ρ k−1−→and then bringing row j down.ρ k−1 ↔ρ k−2−→.⎟⎠ρ j+1 ↔ρ j· · · −→ρ j+1 ↔ρ j+2 ρ j+2 ↔ρ j+3 ρ k−1 ↔ρ k−→ −→ · · · −→⎛⎜⎝⎜⎝.ι φ(k)ι φ(j+1)ι φ(j+2).ι φ(j).ι φ(j+1).ι φ(k−1)Each of these adjacent swaps changes the number of inversions from odd to evenor from even to odd. There are an odd number (k − j) + (k − j − 1) of them.The total change in the number of inversions is from even to odd or from odd toeven.QED4.4 Corollary If a permutation matrix has an odd number of inversions thenswapping it to the identity takes an odd number of swaps. If it has an evennumber of inversions then swapping to the identity takes an even number ofswaps..⎞⎟⎠⎟⎠