Linear Algebra

308 Chapter Four. DeterminantsIf the rows are not adjacent then they can be swapped via a sequence ofadjacent swaps, first bringing row k up⎛⎜⎝..ι φ(j)ι φ(j+1)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(k)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 oddto even.QED4.4 Definition The signum of a permutation sgn(φ) is +1 if the number ofinversions in P φ is even, and is −1 if the number of inversions is odd.4.5 Example With the subscripts from Example 3.8 for the 3-permutations,sgn(φ 1 ) = 1 while sgn(φ 2 ) = −1.4.6 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.Proof. The identity matrix has zero inversions. To change an odd number tozero requires an odd number of swaps, and to change an even number to zerorequires an even number of swaps.QEDWe still have not shown that the permutation expansion is well-defined becausewe have not considered row operations on permutation matrices other than..⎞⎟⎠⎞⎟⎠