Page 52 - Matrices theory and applications

2.9. Exercises
(a) Show that for every i = j, the polynomial X j − X i divides ∆.
(b) Deduce that

(X j − X i ),
∆= a
i<j
where a ∈ K.
(c) Determine the value of a by considering the monomial 35
n
j
X .
j
j=1
(d) Redo this analysis for the matrix
 
X 1 p 1 ··· X n p 1
. .
. .  ,
 
 . .
X p n ··· X p n
1 n
where p 1 ,... ,p n are nonnegative integers.
20. Deduce from the previous exercise that the determinant of the
Vandermonde matrix
1 ··· 1
 
a 1 a n
 ··· 
 2 2 
a a
1 n  , a 1 ,... ,a n ∈ K,
 ··· 

. .
. .
 
 . . 
a n−1 ··· a n−1
1 n
is zero if and only if at least two of the a j ’s coincide.
21. A matrix A ∈ M n (IR) is called a totally positive matrix when all
minors

i 1 i 2 ··· i p
A
j 1 j 2 ··· j p
with 1 ≤ p ≤ n,1 ≤ i 1 < ··· <i p ≤ n and 1 ≤ j 1 < ··· <j p ≤ n are
positive.
(a) Prove that the product of totally positive matrices is totally
positive.
(b) Prove that a totally positive matrix admits an LU factorization
(see Chapter 8), and that every “nontrivial” minor of L and U
is positive. Here, “nontrivial” means

i 1 i 2 ··· i p
L
j 1 j 2 ··· j p
with 1 ≤ p ≤ n,1 ≤ i 1 < ··· <i p ≤ n,1 ≤ j 1 < ··· <j p ≤ l,
and i s ≥ j s for every s.For U,read i s ≤ j s instead. Note:One
says that L and U are triangular totally positive.