4.9 An Identity Of Cauchy
There is a very interesting identity for determinants due to Cauchy.
Theorem 4.9.1 The following identity holds.
Proof: What is the exponent of a2 on the right? It occurs in
Therefore, there are exactly n −
1 factors which contain a2.
has an exponent of n −
Similarly, each ak
is raised to the n −
and the same holds for the bk
as well. Therefore, the right side of 4.9.31
is of the
where c is some constant. Now consider the left side of 4.9.31.
This is of the form
For a given i1
where you can assume the ik are all distinct and the jk are also all distinct because
otherwise sgn will produce a 0. Therefore, in
there are exactly n− 1 factors which contain ak for each k and similarly, there are exactly
n − 1 factors which contain bk for each k. Therefore, the left side of 4.9.31 is of the
and it remains to verify that c = d. Using the properties of determinants, the left side of
4.9.31 is of the form
Let ak →−bk. Then this converges to ∏
The right side of 4.9.31
Therefore, d = c and this proves the identity.