Page 76 - Matrices theory and applications

3.5. Exercises
59
17. Let B ∈ GL n (CC). Verify that the inverse and the Hermitian adjoint
−1
of B
∗
B are similar. Conversely, let A ∈ GL n (CC)be a matrix whose
−1
−1
.
inverse and the Hermitian adjoint are similar: A = PA
P
∗
(a) Show that there exists an invertible Hermitian matrix H such
that H = A HA. Look for an H as a linear combination of P
∗
∗
and of P .
(b) Show that there exists a matrix B ∈ GL n (CC) such that A =
∗
B −1 B .Lookfor a B of the form (aI n + bA )H.
∗
18. Let A ∈ M n (CC) be given, and let λ 1 ,... ,λ n be its eigenvalues. Show,
by induction on n,that A is normal if and only if
n
2 2
|a ij | = |λ l | .
i,j 1
Hint: The left-hand side (whose square root is called Schur’s norm)
is invariant under conjugation by a unitary matrix. It is then enough
to restrict attention to the case of a triangular matrix.
19. (a) Show that | det(I n + A)|≥ 1 for every skew-Hermitian matrix
A, and that equality holds only if A =0 n.
∗
(b) Deduce that for every M ∈ M n (CC)suchthat H := (M +M )/2
is positive definite,
det H ≤| det M|
by showing that H −1 (M − M ) is similar to a skew-Hermitian
∗
matrix. You may use the square root defined at Chapter 7.
20. Describe every positive semidefinite matrix M ∈ Sym (IR) such that
n
m jj =1 for every j and possessing the eigenvalue λ = n (first show
that M has rank one).
21. If A, B ∈ M n×m (CC), define the Hadamard product of A and B by
A ◦ B := (a ij b ij ) 1≤i≤n,1≤j≤m .
(a) Let A, B be two Hermitian matrices. Verify that A ◦ B is
Hermitian.
(b) Assume that A and B are positive semidefinite, of respective
ranks p and q. Using Proposition 3.2.1, show that there exist pq
vectors z αβ such that

A ◦ B = z αβ z ∗ αβ .
α,β
Deduce that A ◦ B is positive semi-definite.
(c) If A and B are positive definite, show that A ◦ B also is positive
definite.