Lemma 2.1. Let A = (a
ij
)n × n, B = (b
ij
)n × n(n = 2m) be two centrosymmetric matrices with the following form:
(2.1)
Then, A ○ B is a centrosymmetric matrix.
Proof. By the definition of Hadamard product,
(2.2)
We shall prove the following
(2.3)
From (2.1),
(2.4)
and
(2.5)
Since
(2.6)
we have the following
(2.7)
From (2.5) and (2.7), it is clear that
(2.8)
Similarly, we can prove that
(2.9)
From (2.8) and (2.9), we can see that (2.3) holds. By Lemma 1.1, A ○ B is a centrosymmetric matrix.
Lemma 2.2 (see [4]) Let A = (a
ij
)n × n(n = 2m) a positive semidefinite centrosymmetric matrix with the following form
Let M = B - J
m
C and N = B + J
m
C. Then, M, N are positive semidefinite matrices.
Theorem 2.1 Let A, B ∈ ℂn × n(n = 2m) be two positive semidefinite centrosymmetric matrices with the same form as in (2.1). Let
and
Then, the following inequality holds:
(2.10)
Proof. Since A, B are centrosymmetric matrices, by Lemma 2.1, A ○ B is centrosymmetric and
From Lemma 1.1,
P is orthogonal, then
and
Similarly from Lemma 1.1,
Then,
and
Since both A and B are positive semidefinite matrices, by Lemma 1.2
Thus,
Theorem 2.2. Let A ∈ ℂn × n(n = 2m) be postive semidefinite centroysymmetric with the form:
Let M = B - J
m
C, N = B + J
m
C. Then, for positive integers s, m, t, the following two equalities hold
(2.11)
and
(2.12)
Proof. By Lemma 1.1,
Since A is positive semidefinite, we have
Thus,
From Lemma 2.2, M and N are positive semidefinite. Combining Lemma 1.3, we have
Theorem 2.3 Let A, B ∈ ℂn × n(n = 2m) are positive semidefinite centrosymmetric matrices with the same form as in (2.1). Let
Then, the following inequality holds
(2.13)
Proof. From Lemma 1.1, there exists an orthogonal matrix P such that
According to Definition 1.1,
From Lemma 1.4,
Since P is orthogonal, we have tr(PTABP)m= tr(AB)m. By Lemma 1.4, the following holds