Page 309 - DMTH502_LINEAR_ALGEBRA

Unit 28: Positive Forms and More on Forms

Proof: Since (A*) = k ( ) (1 k n ), the principal minors of A are all different from 0. Hence, Notes
A
k
by the lemma used in the proof of Theorem 2, there exists an upper-triangular matrix P with
P = 1 such that A*P is lower-triangular. Therefore, P*A = (A*P)* is upper-triangular. Since the
kk
product of two upper-triangular matrices is again upper triangular, it follows that P*AP is
upper-triangular. This shows the existence but not the uniqueness of P. However, there is another
more geometric argument which may be used to prove both the existence and uniqueness of P.
Let W be the subspace spanned by , ..., and W’ the set of all in V such that f( , ) = 0 for
k 1 k k
every in W . Since (A) 0, the k k matrix M with entries
k k
M = f( , ) = A
ij j i ij
(1 i, j k) is invertible. By Theorem 3

V = W W’ .
k k
Let E be the projection of V on W which is determined by this decomposition, and set E = 0. Let
k k 0
= – E , (1 k n)
k k k –1 k
Then = , and E belongs to W for k > 1. Thus when k > 1, there exist unique scalars P such
1 1 k–1 k k–1 jk
that
k 1
E = – P jk j
k –1 k
j 1
Setting P = 1 and P = 0 for j < k, we then have an n × n upper triangular matrix P with P = 1 and
kk jk kk
k
B = P jk j
k
j 1
for k =1, ..., n. Suppose 1 i k. Then B is in W W since B belongs to W’ , it follows that
k i k–1 k k –1
f( , ) = 0. Let B denote the matrix of f in the ordered basis ( , ... ). Then
i k 1 n
B = f( , )
ki i k
so B = 0 when k > i. Thus B is upper-triangular. On the other hand,
ki
B = P*AP.

Self Assessment

1. Which of the following matrices are positive?

1 1 1 1 1 2 1 3
1 2 1 1 i
, , 2 1 1 , 1 2 1 2 1 4
3 4 1 i 3
3 1 1 1 3 1 4 1 5
2. Prove that the product of two positive linear operators is positive if and only if they
commute.

3. Let S and T be positive operators. Prove that every characteristic value of ST is positive.

28.3 Summary

 In this unit we are studying the form f on a finite vector space being non-negative.
 We obtain certain equivalent properties and show that when the matrix A of linear operator
is Hermitian i.e. A + A* as well as the principal minors of the matrix A are all positive.

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