Page 77 - Matrices theory and applications

3. Matrices with Real or Complex Entries
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(d) Construct an example for which p, q < n,but A ◦ B is positive
definite.
22. (Fiedler and Pt´k [13]) Given a matrix A ∈ M n (IR), we wish to prove
a
the equivalence of the following properties:
P1 For every vector x = 0 there exists an index k such that
x k (Ax) k > 0.
P2 For every vector x = 0 there exists a diagonal matrix D with
positive diagonal elements such that the scalar product (Ax, Dx)
is positive.
P3 For every vector x = 0 there exists a diagonal matrix D with
nonnegative diagonal elements such that the scalar product
(Ax, Dx) is positive.
P4 The real eigenvalues of all principal submatrices of A are positive.
P5 All principal minors of A are positive.
n
We shall use the following notation: if x ∈ IR and if J is the index set
k
J
of its nonzero components, then x denotes the vector in IR ,and k
the cardinality of J, where one retains only the nonzero components
J
of x.To the set J one also associates the matrix A , retaining only
the indices in J.
(a) Prove that Pj implies P(j+1) for every j =1,... , 4.
(b) Assume P5. Show that for every diagonal matrix D with
nonnegative entries, one has det(A + D) > 0.
(c) Then prove that P5 implies P1.