Linear Algebra

308 Chapter 4. Determinants
To state this as a formula, we introduce a notation for permutation matrices.
Let ιj be the row vector that is all zeroes except for a one in its j-th entry, so
that the four-wide ι2 is � 0 1 0 0 � . We can construct permutation matrices
by permuting — that is, scrambling — the numbers 1, 2, ... , n, and using them
as indices on the ι’s. For instance, to get a 4×4 permutation matrix matrix, we
can scramble the numbers from 1 to 4 into this sequence 〈3, 2, 1, 4〉 and take the
corresponding row vector ι’s.
⎛ ⎞
ι3
⎜ι2⎟
⎜ ⎟
⎝ι1⎠
=
⎛ ⎞
0 0 1 0
⎜
⎜0
1 0 0 ⎟
⎝1
0 0 0⎠
0 0 0 1
ι4
3.7 Definition An n-permutation is a sequence consisting of an arrangement
of the numbers 1, 2, ... , n.
3.8 Example The 2-permutations are φ1 = 〈1, 2〉 and φ2 = 〈2, 1〉. These are
the associated permutation matrices.
Pφ1 =
� � � �
ι1 1 0
=
Pφ2 0 1
=
� � � �
ι2 0 1
=
1 0
ι2
We sometimes write permutations as functions, e.g., φ2(1) = 2, and φ2(2) = 1.
Then the rows of Pφ2 are ι φ2(1) = ι2 and ι φ2(2) = ι1.
The 3-permutations are φ1 = 〈1, 2, 3〉, φ2 = 〈1, 3, 2〉, φ3 = 〈2, 1, 3〉, φ4 =
〈2, 3, 1〉, φ5 = 〈3, 1, 2〉, andφ6 = 〈3, 2, 1〉. Here are two of the associated permu-
tation matrices.
Pφ2 =
⎛
⎝
ι1
ι3
ι2
⎞ ⎛ ⎞
1 0 0
⎠ = ⎝0
0 1⎠
Pφ5
0 1 0
=
⎛
⎝
ι1
ι3
ι1
ι2
⎞ ⎛
0
⎠ = ⎝1
0
0
⎞
1
0⎠
0 1 0
For instance, the rows of Pφ5 are ι φ5(1) = ι3, ι φ5(2) = ι1, andι φ5(3) = ι2.
3.9 Definition The permutation expansion for determinants is
�
�t1,1
�
�t2,1
�
�
�
�
�
t1,2
t2,2
.
...
...
�
t1,n�
�
t2,n�
�
�
�
�
�
tn,1 tn,2 ... tn,n
where φ1,... ,φk are all of the n-permutations.
= t 1,φ1(1)t 2,φ1(2) ···t n,φ1(n)|Pφ1 |
+ t 1,φ2(1)t 2,φ2(2) ···t n,φ2(n)|Pφ2 |
.
+ t 1,φk(1)t 2,φk(2) ···t n,φk(n)|Pφk |
This formula is often written in summation notation
|T | =
�
permutations φ
t 1,φ(1)t 2,φ(2) ···t n,φ(n) |Pφ|