Page 53 - Matrices theory and applications

2. Square Matrices
36
(c) Show that a Vandermonde matrix (see the previous exercise) is
totally positive whenever 0 <a 1 < ··· <a n .
22. Multiplying a Vandermonde matrix by its transpose, show that

.
.
.

 n s 1 ··· . s n−1 
. .
 s 1 s 2  2
det   = (a j − a i ) ,
. .
. . .
 . 
 . . .  i<j
s n−1 ··· ··· s 2n−2
q q
where s q := a + ··· + a .
1 n
23. The discriminant of a matrix A ∈ M n (k)is the number
2
d(A):= (λ j − λ i ) ,
i<j
where λ 1 ,... ,λ n are the eigenvalues of A, counted with multiplicity.
(a) Verify that the polynomial
2
∆(X 1 ,... ,X n ):= (X j − X i )
i<j
is symmetric. Therefore, there exists a unique polynomial Q ∈
ZZ[Y 1 ,... ,Y n ]such that
∆= Q(σ 1 ,... ,σ n ),
where the σ j ’s are the elementary symmetric polynomials

σ 1 = X 1 + ··· + X n ,... ,σ n = X 1 ··· X n .

(b) Deduce that there exits a polynomial D ∈ ZZ[x ij ] in the indeter-
minates x ij ,1 ≤ i, j ≤ n, such that for every k and every square
matrix A,

d(A)= D(a 11 ,a 12 ,... ,a nn ).

(c) Consider the restriction D S of the discriminant to symmetric
matrices, where x ji is replaced by x ij whenever i< j.Prove that
D S takes only nonnegative values on IR n(n+1)/2 . Show, however,
that D S is not the square of a polynomial if n ≥ 2 (consider first
the case n =2).