Linear Algebra, Theory And Applications, 2012a

84 DETERMINANTS
3.3.1 The Function sgn
The following Lemma will be essential in the definition of the determinant.
Lemma 3.3.1 There exists a unique function, sgn n which maps each ordered list of numbers
from {1, ··· ,n} to one of the three numbers, 0, 1, or −1 which also has the following
properties.
sgn n (1, ··· ,n) = 1 (3.2)
sgn n (i 1 , ··· ,p,··· ,q,··· ,i n )=− sgn n (i 1 , ··· ,q,··· ,p,··· ,i n ) (3.3)
In words, the second property states that if two of the numbers are switched, the value of the
function is multiplied by −1. Also, in the case where n>1 and {i 1 , ··· ,i n } = {1, ··· ,n} so
that every number from {1, ··· ,n} appears in the ordered list, (i 1 , ··· ,i n ) ,
sgn n (i 1 , ··· ,i θ−1 ,n,i θ+1 , ··· ,i n ) ≡
(−1) n−θ sgn n−1 (i 1 , ··· ,i θ−1 ,i θ+1 , ··· ,i n ) (3.4)
where n = i θ in the ordered list, (i 1 , ··· ,i n ) .
Proof: To begin with, it is necessary to show the existence of such a function. This is
clearly true if n =1. Define sgn 1 (1) ≡ 1 and observe that it works. No switching is possible.
Inthecasewheren =2, it is also clearly true. Let sgn 2 (1, 2) = 1 and sgn 2 (2, 1) = −1
while sgn 2 (2, 2) = sgn 2 (1, 1) = 0 and verify it works. Assuming such a function exists for n,
sgn n+1 will be defined in terms of sgn n . If there are any repeated numbers in (i 1 , ··· ,i n+1 ) ,
sgn n+1 (i 1 , ··· ,i n+1 ) ≡ 0. If there are no repeats, then n + 1 appears somewhere in the
ordered list. Let θ be the position of the number n + 1 in the list. Thus, the list is of the
form (i 1 , ··· ,i θ−1 ,n+1,i θ+1 , ··· ,i n+1 ) . From(3.4)itmustbethat
sgn n+1 (i 1 , ··· ,i θ−1 ,n+1,i θ+1 , ··· ,i n+1 ) ≡
(−1) n+1−θ sgn n (i 1 , ··· ,i θ−1 ,i θ+1 , ··· ,i n+1 ) .
It is necessary to verify this satisfies (3.2) and (3.3) with n replaced with n +1. The first of
these is obviously true because
sgn n+1 (1, ··· ,n,n+1)≡ (−1) n+1−(n+1) sgn n (1, ··· ,n)=1.
If there are repeated numbers in (i 1 , ··· ,i n+1 ) , then it is obvious (3.3) holds because both
sides would equal zero from the above definition. It remains to verify (3.3) in the case where
there are no numbers repeated in (i 1 , ··· ,i n+1 ) . Consider
(
)
sgn n+1 i 1 , ··· , p, r ··· , q, s ··· ,i n+1 ,
where the r above the p indicates the number p is in the r th position and the s above the q
indicates that the number, q is in the s th position. Suppose first that r