Chapter 28

The Mathematical Theory OfDeterminants∗

28.0.1 The Function sgn

The following Lemma will be essential in the definition of the determinant.

Lemma 28.0.1 There exists a function, sgnn 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.

sgnn (1, · · · ,n) = 1 (28.1)

sgnn (i1, · · · , p, · · · ,q, · · · , in) =−sgnn (i1, · · · ,q, · · · , p, · · · , in) (28.2)

In words, the second property states that if two of the numbers are switched, the value ofthe function is multiplied by−1. Also, in the case where n > 1 and {i1, · · · , in}= {1, · · · ,n}so that every number from {1, · · · ,n} appears in the ordered list, (i1, · · · , in) ,

sgnn (i1, · · · , iθ−1,n, iθ+1, · · · , in)≡

(−1)n−θ sgnn−1 (i1, · · · , iθ−1, iθ+1, · · · , in) (28.3)

where n = iθ in the ordered list, (i1, · · · , in) .

Proof: Define sign(x) = 1 if x > 0,−1 if x < 0 and 0 if x = 0. If n = 1, there is onlyone list and it is just the number 1. Thus one can define sgn1 (1)≡ 1. For the general casewhere n > 1, simply define

sgnn (i1, · · · , in)≡ sign

(∏r<s

(is− ir)

)This delivers either −1,1, or 0 by definition. What about the other claims? Suppose youswitch ip with iq where p < q so two numbers in the ordered list (i1, · · · , in) are switched.

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