Some inequalities for sums of nonnegative definite matrices in quaternions
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Some matrix versions of the Cauchy-Schwarz and Frucht-Kantorovich inequalities are established over the quaternionic algebra. As applications, a group of inequalities for sums of Hermitian nonnegative definite matrices over the quaternionic algebra are derived. Let a = a0 + a1 i + a2 j + a3 k be a quaternion, where a0 ,...,a3 are numbers from the real field R and the three imaginary units i, j, and k satisfy i2 = j 2 = k2 = −1,
i j = − ji = k,
jk = −k j = i,
ki = −ik = j.
(1)
The collection of all quaternions is denoted by H and is called the real quaternionic algebra. This algebra was first introduced by Hamilton in 1843 (see [5, 6]), and is often called the Hamilton quaternionic algebra. It is well known that H is an associative division algebra over R. For any a = a0 + a1 i + a2 j + a3 k ∈ H, the conjugate of a = a0 + a1 i + a2 j + a3 k is defined to be a = a0 − a1 i − a2 j − a3 k, which satisfies a = a,
a + b = a + b,
ab = ba √
(2)
√
for all a,b ∈ H. The norm of a is defined to be |a| = aa = aa = a20 + a21 + a22 + a23 . Let A = (ast ) be an m × n matrix over H, where ast ∈ H. The conjugate transpose of A is defined to be A∗ = (ats ). A square matrix A over H is called Hermitian if A∗ = A. General properties of matrices over H can be found in [13, 18]. Because H is noncommutative, one cannot directly extend various results on complex numbers to quaternions. On the other hand, H is known to be algebraically isomorphic to the two matrix algebras consisting of
ψ(a) = def
a0 + a1 i a2 − a3 i
− a2 + a3 i ∈ C2×2 , a0 − a1 i
a0 a def 1 φ(a) = a2 a3
Copyright © 2005 Hindawi Publishing Corporation Journal of Inequalities and Applications 2005:5 (2005) 449–458 DOI: 10.1155/JIA.2005.449
−a1
a0 a3 −a2
−a2 −a3
a0 a1
−a3
a2 ∈ R4×4 , −a1 a0 (3)
450
Inequalities for matrices
respectively. Moreover, it is shown in [13] that the diagonal matrix diag(a,a) satisfies the following universal similarity factorization equality (USFE): P diag(a,a)P ∗ = ψ(a),
(4)
where
−i
1 1 P= √ − j 2
(5)
k
is a unitary matrix over H, that is, PP ∗ = P ∗ P = I2 ; the diagonal matrix diag(a,a,a,a) satisfies the following USFE: Q diag(a,a,a,a)Q∗ = φ(a),
(6)
where the matrix Q has the following independent expression:
1
1 −i Q = Q∗ = 2 − j −k
i 1 −k j
j k k − j , 1 i −i 1
(7)
which is a unitary matrix over H. The two equalities in (4) and (6) reveal two fundamental facts that the quaternion a is an eigenvalue of multiplicity two for the complex matrix ψ(a) and an eigenvalue of multiplicity four for the real matrix φ(a). In general, for any m × n matrix A = A0 + A1 i + A2 j + A3 k ∈ Hm×n , where A0 ,...,A3 ∈ m R ×n , the block-diagonal matrix diag(A,A) satisfies the following universal factorization equality:
A0 + A1 i − A2 + A3 i P2m diag(A,A)P2n = A2 − A3 i A0 − A1 i
def
∗
= Ψ(A) ∈ C2m×2n ,
(8)
∗ where P2m and P2n are the following two unitary matrices over H:
Im 1 2 − jIm
P2m = √
−iIm
kIm
,
∗
1 In 2 iIn
P2n = √
jIn . −kIn
(9
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