Page 72 - Matrices theory and applications

3.4.1 Hadamard’s Inequality
Proposition 3.4.2 Let H ∈ H n be a positive semidefinite Hermitian
matrix. Then
n

h jj .
det H ≤
j=1
If H ∈ HPD n , the equality holds only if H is diagonal. 3.5. Exercises 55
Proof
If det H = 0, there is nothing to prove, because the h jj are nonnegative
j
(these are numbers (e ) He ). Otherwise, H is positive definite and one
j ∗
has h jj > 0. We restrict attention to the case with a constant diagonal
−1/2 −1/2
by letting D := diag(h 11 ,... ,h nn ) and writing (det H)/( j h jj )=
det DHD =det H , where the diagonal entries of H equal one. There

remains to prove that det H ≤ 1. However, the eigenvalues µ 1 ,... ,µ n of

H are strictly positive, of sum n. Since the logarithm is concave, one has
1 1 1

log det H = log µ j ≤ log µ j =log 1 =0,
n n n
j
which proves the inequality. Since the concavity is strict, the equality holds
only if µ 1 = ··· = µ n = 1, but then H is similar, thus equal to I n .In that

case, H is diagonal.
Applying proposition 3.4.2 to matrices of the form M M or MM ,one
∗
∗
obtains the following result.
Theorem 3.4.3 For M ∈ M n (CC), one has
1/2
  1/2
n n n n
2 2

| det M|≤  |m ij |  , | det M|≤ |m ij | .
i=1 j=1 j=1 i=1
When M ∈ GL n (CC), the first (respectively the second) inequality is an
equality only if the rows (respectively the columns) of M are pairwise
orthogonal.
3.5 Exercises
1. Show that the eigenvalues of skew-Hermitian matrices, or as well
those of real skew-symmetric matrices, are pure imaginary.
2. Let P, Q ∈ M n (IR) be given. Assume that P + iQ ∈ GL n (CC). Show
that there exist a, b ∈ IR such that aP + bQ ∈ GL n (IR). Deduce that
if M, N ∈ M n (IR) are similar in M n (CC), then these matrices are
similar in M n (IR).