Linear Algebra, 2020a

Section I. Definition 347
could be computed with one swap
or with three.
ρ 1 ↔ρ 2
P φ −→
ρ 3 ↔ρ 1 ρ 2 ↔ρ 3 ρ 1 ↔ρ 3
P φ −→ −→ −→
⎛
⎞
1 0 0 0
0 1 0 0
⎜
⎟
⎝0 0 1 0⎠
0 0 0 1
⎛
⎞
1 0 0 0
0 1 0 0
⎜
⎟
⎝0 0 1 0⎠
0 0 0 1
Both reductions have an odd number of swaps so in this case we figure that
|P φ | =−1 but if there were some way to do it with an even number of swaps then
we would have the determinant giving two different outputs from a single input.
Below, Corollary 4.5 proves that this cannot happen — there is no permutation
matrix that can be row-swapped to an identity matrix in two ways, one with an
even number of swaps and the other with an odd number of swaps.
4.1 Definition In a permutation φ = 〈...,k,...,j,...〉, elements such that k>j
are in an inversion of their natural order. Similarly, in a permutation matrix
two rows
⎛ ⎞
.
ι ḳ P φ =
. ⎜ι ⎝ j... ⎟
⎠
such that k>jare in an inversion.
4.2 Example This permutation matrix
⎛
⎞ ⎛ ⎞
1 0 0 0 ι 1
0 0 1 0
⎜
⎟
⎝0 1 0 0⎠ = ι 3
⎜ ⎟
⎝ι 2 ⎠
0 0 0 1 ι 4
has a single inversion, that ι 3 precedes ι 2 .
4.3 Example There are three inversions here:
⎛
⎜
0 0 1
⎞ ⎛
⎟ ⎜
ι ⎞
3
⎟
⎝0 1 0⎠ = ⎝ι 2 ⎠
1 0 0 ι 1