Selçuk J. Appl. Math. Selçuk Journal of Special Issue. pp. 87-94, 20100 Applied Mathematics
Inequalities for Quaternion Matrices Zübeyde Ulukök, Ramazan Türkmen
Selçuk University, Science Faculty, Department of Mathematics, 42003, Kampus, Konya, Türkiye
e-mail: zulukok@ selcuk.edu.tr,rturkm en@ selcuk.edu.tr
Presented in 2National Workshop of Konya Ere˘gli Kemal Akman College, 13-14 May 2010.
Abstract. In this paper, we present the trace inequalities for the product of powers of quaternion positive semidefinite matrices by handing some inequalities which previously obtained any complex matrices.
Key words: Quaternions, Trace Inequalities.
2000 Mathematics Subject Classification: 15A45, 15A66. 1. Introduction
Quaternions were introduced by the Irish mathematician Sir William Rowan Hamilton (1805-1865) in 1843 as he looked for ways of extending complex num-bers to higher spatial dimensions.There has been an increasing interest in al-gebra problems on quaternion field since many alal-gebra problems on quaternion field were encountered in some applied sciences, such as the quantum physics, geostatics, the figure and pattern recognition and the space telemetry (see, e.g., [2], [3]). Until now, the determinant, eigenvalues and the system of quaternion matrix equation have been investigated widely. Some authors have been shown some inequalities which have given for real or complex matrices for quaternion matrices ([1], [7], [8], [9], [10]).
We use R C and H to denote the set of real numbers, the set of complex numbers, and the set of quaterninons, respectively.
For any ∈ H, we have the unique representation = 1 + + + , where {1 } is the basis of H. It is well-known that is the multiplicative identity of H, and 12 = 2 = 2 = 2 = −1, = = = and = − = − = − .
For each = 1 + + + ∈ Hdefine the conjugate of by = 1 − − − .
Obviously we have = = 2+ 2+ 2+ 2. This implies that = = 0 if and only if = 0. So is invertible in H if 6= 0.
We note that as subalgebras of H, the meaning of conjugate in R, or C is as usual (for any ∈ R we have = ).
We can consider R and C as real subalgebras of H : R = {1 : ∈ R}, and C = {1 + : ∈ R}.
We define the real representation of H, i.e., : H → M4(R) by
() = (1 + + + ) = ⎛ ⎜ ⎜ ⎝ − − − − − − ⎞ ⎟ ⎟ ⎠ where = 1 + + + ∈ H. Note that () is the transpose of ().
From the real representation of H, we define a faithful representation by : (H) →4(R) as follows:
() = ³[]=1
´
=³[ ()]=1
´
for all matrices = []=1 ∈ (H). (H+) and (H∗) denote the
set of quaternion positive semidefinite matrices and self-conjugate quaternion matrices, respectively.
We note that each is an injective and homomorphism; and for all ∈ (H)
(∗) = ()∗.
The authors have given some inequalities for quaternion matrices. For example , Thompson [9] extended the complex matrix-valued triangle inequality to qua-trenion matrices: for any × quaternion matrices and , there are × unitary quaternion matrices and such that
| + | ≤ || ∗+ || ∗.
Zheng [4] have proved Young’s inequality in quaternion matrices: for any × quaternion matrices and any ∈ (1 ∞) with 1
+ 1
= 1 there exists
× unitary quaternion matrix such that |∗| ≤ 1 || +1 ||
Furthermore, Hu [1] has given some trace inequalities for the product of quater-nion positive definite matrices: for quaterquater-nion positive definite matrices and such that = p () ≤p () () ≤ () + ()2 ≤ r 2() + 2() 2
and () ≤ ( ())1( ()) 1 where 1 + 1 = 1.
In this study, we give some trace inequalities for the product of powers of two quaternion positive semidefinite matrices.
2. Definitions and Lemmas
Definition 1. Let ∈ (H∗). is said to be the self-conjugate quaternion
matrix if ∗= .
Definition 2. Let ∈ (H). is said to be the quaternion unitary matrix
if ∗ = ∗= .(H) is the set of quaternion unitary matrices.
Definition3. Let ∈ (H).
X
=1
is said to be the trace of matrix ,
remarked by (). That is () =
X
=1
.
Lemma 1. [1] Let ∈ (H∗) and ∈ (H∗). If there exists ∈
(H), such that = ∗, then, () = ().
Lemma 2. [10] Let ∈ (H), then ∗ is nonnegative definite.
Further-more, if ∈ (H) is nonnegative definite, then there are matrices ∈
(H) such that
(i) is unitary and is diagonal matrix with nonnegative diagonal entries 1 2 ;
(ii) ∗ = ;
(iii) () = {1 2 };
(iv) If ∈ () appears times on the diagonal of , then the geometric
multiplicity of as an eigenvalue of () is 4.
Lemma 3. [9] For any ∈ (H),
(i) (||) = |()|;
(ii) (||) = |()| for any nonnegative definite ; (iii) (||) = |() ()|.
The meaning of || is that || = (∗)12.
Lemma 4. [6] Let ∈ (C) then X =1 ¯ ¯(()2) ¯ ¯ ≤ X =1 ((∗∗)) ≤ X =1 ((∗)(∗)) 1 ≤ ≤ ∈ N
Lemma 5. [6] Let ∈ (C), then
¯
¯()2¯¯ ≤ (∗∗)
≤ (∗)(∗) ∈ N. 3. Main Results
Theorem 1. Let ∈ (H+) ∈ (H+). If are commutative, then
()≤©22ª
1 2
where is a positive integer.
Proof. We know that if ∈ (C) and ∈ (C) are positive semidefinite
matrices; then we have (see, e.g., [5])
(1) ()≤n ()2 ()2o
1 2
where is an integer. If ∈ (H+) and ∈ (H+) then () ∈
4(R) and () ∈ 4(R) are positive semidefinite matrices. For () ≥
0 there exists a unique (())
1 2 ≥ 0.Thus, 0 ≤ ³12 () () 1 2 () ´ = (() ())
Also, we know that for positive semidefinite matrices , is positive semidefinite if and only if = . Then, () () = () is pos-itive semidefinite matrices. According to Lemma 2 is quaternion pospos-itive semidefinite matrix. Since is a homomorphism, we write
(() ()) = (()) = 4 X =1 (()) = 4 X =1 (())
Since is a homomorphism and from inequality (1)
(()) = [() ()]≤n (())2 (())2o 1 2 [(() )] ≤ ©¡ ¡ 2¢¢¡ ¡ 2¢¢ª 1 2 4 X =1 [(() )] ≤ (4 X =1 £ ¡ 2¢¤ 4 X =1 £ ¡ 2¢¤ )1 2
From Lemma 2, ∈ () appears times on the diagonal of , then the
geometric multiplicity of as an eigenvalue of () is 4. That is, if is a
multiple eigenvalues of a matrix then is four multiple of matrix ().By
means of this expression, we write
X =1 [()] ≤ ( X =1 £¡ 2¢¤ X =1 £¡ 2¢¤ )1 2 Therefore, we get ()≤n ()2 ()2o 1 2 .
Theorem 2. Let ∈ (H+) ∈ (H+). If are commutative, then
()2≤ ¡22¢≤ ¡22¢
where is a positive integer.
Proof. In Lemma 5, if ∈ (C) are positive semidefinite matrices, then
()2≤ ¡22¢≤ ¡22¢
If ∈ (H+) and ∈ (H+) then () ∈ 4(R) and () ∈
4(R) are positive semidefinite matrices.
0 ≤ [() ()] 2 = [()] 2 = h ³ ()2´i ≤ h(())2(())2i= £¡2¢¡2¢¤ = £¡22¢¤= h³¡22¢´i≤ [()]2[()]2 = £¡2¢¡2¢¤= £¡22¢¤
From the definition of trace, we write
4 X =1 h ³()2´i≤ 4 X =1 h ³¡22¢´i≤ 4 X =1 £ ¡ 22¢¤. According to Lemma 2, if is a multiple eigenvalues of a matrix then is
four multiple of matrix () Then, we have
X =1 h ()2i≤ X =1 h¡ 22¢i≤ X =1 £ 22¤ That is, ()2≤ ¡22¢≤ ¡22¢.
Theorem 3. Let ∈ (H+) ∈ (H+). If are commutative, then
()≤ () where is a positive real numbers.
Proof. If ∈ (C) are positive semidefinite matrices, we know that
()≤ () .
We want to extend this inequality to quaternion positive semidefinite matrices. If ∈ (H+) and ∈ (H+) since () ∈ 4(R) and () ∈
4(R) are positive semidefinite matrices we write
0 ≤ [() ()] = [()] = [(())] ≤ [(())(())] = [() ()] = [()] Thus, because of () ≥ 0 we have
4 X =1 [(() )] ≤ 4 X =1 [()]
According to Lemma 2, we get
X =1 [()] ≤ X =1 [] .
Therefore the desired is obtained. Theorem 4. Let =
µ
∗
¶
be quaternion positive semidefinite block matrix such that ∈ (H). Then
(||) ≤ { () ()}2 where 0. Proof. If µ ∗ ¶
is positive semidefinte block matrices such that ∈ (C), then (see, e.g, [11])
where 0. If = µ
∗
¶
is positive semidefinite quaternion matrices, then () = µ ∗ ¶ = µ () () (∗) () ¶ ∈ 8(R)
for homomorphism . Since matrix is positive semidefinite then ()
is positive semidefinite. As () is a matrix with real entries, according to
inequality (2) we write (|()| ) ≤ { (()) (())}12.
Since is a homomorphism, by using Lemma 3, we have [(||)] ≤ { (()) (())}12. That is, 4 X =1 [(|| )] ≤ (4 X =1 [()] 4 X =1 [()] )1 2 . From Lemma 2, we get
X =1 [||] ≤ ( X =1 [] X =1 [] )1 2 Thus the proof is completed.
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