Linear Algebra, Theory And Applications, 2012a

104 DETERMINANTS
13. Let U be an open set in R n and let g :U → R n be such that all the first partial
derivatives of all components of g exist and are continuous. Under these conditions
form the matrix Dg (x) givenby
Dg (x) ij
≡ ∂g i (x)
≡ g i,j (x)
∂x j
The best kept secret in calculus courses is that the linear transformation determined
by this matrix Dg (x) is called the derivative of g and is the correct generalization
of the concept of derivative of a function of one variable. Suppose the second partial
derivatives also exist and are continuous. Then show that
∑
(cof (Dg)) ij,j
=0.
j
Hint: First explain why ∑ i g i,k cof (Dg) ij
= δ jk det (Dg) . Next differentiate with
respect to x j and sum on j using the equality of mixed partial derivatives. Assume
det (Dg) ≠ 0 to prove the identity in this special case. Then explain using Problem 10
why there exists a sequence ε k → 0 such that for g εk (x) ≡ g (x)+ε k x, det (Dg εk ) ≠0
and so the identity holds for g εk . Then take a limit to get the desired result in general.
This is an extremely important identity which has surprising implications. One can
build degree theory on it for example. It also leads to simple proofs of the Brouwer
fixed point theorem from topology.
14. A determinant of the form ∣∣∣∣∣∣∣∣∣∣∣∣∣ 1 1 ··· 1
a 0 a 1 ··· a n
a 2 0 a 2 1 ··· a 2 n
.
.
.
a n−1
0 a n−1
1 ··· a n−1
n
a n 0 a n 1 ··· a n ∣
n
is called a Vandermonde determinant. Show this determinant equals
∏
(a j − a i )
0≤ii.
Hint: Show it works if n = 1 so you are looking at
∣ 1 1 ∣∣∣
∣ a 0 a 1
Then suppose it holds for n − 1 and consider the case n. Consider the polynomial in
t, p (t) which is obtained from the above by replacing the last column with the column
( ) 1 t ··· t
n T
.
Explain why p (a j )=0fori =0, ··· ,n− 1. Explain why
n−1
∏
p (t) =c (t − a i ) .
i=0
Of course c is the coefficient of t n . Find this coefficient from the above description of
p (t) and the induction hypothesis. Then plug in t = a n and observe you have the
formula valid for n.