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On Positive Definite Solutions of Quaternionic Matrix Equations World Academy of Science, Engineering and Technology International Journal of Mathematical and Computational Sciences Vol:4, No:1, 2010 On Positive Definite Solutions of Quaternionic Matrix Equations Minghui Wang H Abstract—The real representation of the quaternionic matrix is A is positive (semi)definite if x Ax > 0(≥ 0) for any nonzero definited and studied. The relations between the positive (semi)define vector x ∈ Qn. quaternionic matrix and its real representation matrix are presented. By means of the real representation, the relation between the positive (semi)definite solutions of quaternionic matrix equations and those of II. REAL REPRESENTATION corresponding real matrix equations is established. In this section, we will give the definition of the real Keywords—Matrix equation, Quaternionic matrix, Real represen- representation and study the relation between the positive tation, positive (semi)definite solutions. (semi)define quaternionic matrix and its real representation matrix. m×n Let Al ∈ R (l =1, 2,,3, 4). The real representation I. INTRODUCTION matrix is defined[7] in the form ⎛ ⎞ N the study of quaternionic quantum mechanics and some A1 −A2 −A3 −A4 ⎜ ⎟ other applications of quaternions [1], [2], [3], one often R ⎜ A2 A1 −A4 A3 ⎟ 4m×4n I A ≡ ⎝ ⎠ ∈ R . (1) encounters the problem of solutions of quaternionic linear A3 A4 A1 −A2 equations. Because of noncommutativity of quaternions, solv- A4 −A3 A2 A1 ing quaternionic linear equations is more difficult. In papers[4], AR [5], [6], by means of a complex representation and a com- The real matrix is uniquely determined by quaternion A = A + A i + A j + A k ∈ Qm×n panion vector, the authors have studied quatemionic linear matrix 1 2 3 4 , and it is said A equations and presented a Cramer rule for quaternionic linear to be a real representation matrix of quaternion matrix . Let It be t × t identity matrix and define equations and an algebraic algorithm for the least squares ⎛ ⎞ problem, respectively, in quaternionic quantum theory. In the It 000 ⎜ ⎟ paper[8], by means of a real representation of the quaternionic ⎜ 0 −It 00⎟ Pt = ⎝ ⎠ , (2) matrix, we gave an iterative algorithms for the least squares 00It 0 problem in quaternionic quantum theory. 000−I ⎛ t ⎞ How to find positive (semi)definite solution of quaternionic 0 −It 00 matrix equations is also an important problem in quaternionic ⎜ ⎟ ⎜ It 000⎟ Qt = ⎝ ⎠ , (3) quantum theory. However, to our best knowledge, the problem 00 0It has not been studied for its difficulty. 00−It 0 In this paper, we will pay attention to positive (semi)definite ⎛ ⎞ 000−It solutions of quaternionic matrix equations by means of a real ⎜ ⎟ ⎜ 00It 0 ⎟ representation of the quaternionic matrix and establish the St = ⎝ ⎠ , (4) 0 −It 00 relation between this problem and the corresponding problem I 000 in the real number field. Because the latter has been studied ⎛ t ⎞ 00−It 0 wildly, we may apply the existing results to the former. ⎜ ⎟ R Q = R⊕Ri⊕Rj⊕Rk ⎜ 00 0 −It ⎟ Let denote the real number field, Rt = . 2 2 2 ⎝ I 00 0⎠ (5) the quaternion field, where i = j = k = −1,ij = −ji = t 0 I 00 k. For any quaternion a = a1 +a2i+a3j +a4k where aj ∈ R, t a a¯ = a1 − a2i − a3j − a4k International Science Index, Mathematical and Computational Sciences Vol:4, No:1, 2010 waset.org/Publication/4376 the conjugate of is . For any Then it is easy verify the following properties. quaternion matrix A, AT , A¯ and AH denote the transpose, Proposition 2.1. Let A, B ∈ Qm×n,C ∈ Qn×s,α∈ R. Then conjugate and conjugate transpose of A over quaternion field, R R R R R m×n (a). (A + B) = A + B , (αA) = αA , respectively. F denotes the set of m × n matrices on a R R R n×n H H (AC) = A C , field F.ForA ∈ Q , A is unitary if A A = AA = I and 2 2 2 T H n×n (b). Qm = Rm = Sm = −I4m,Qm = −Qm, Hermitian if A = A. For any Hermitian matrix A ∈ Q , T T Rm = −Rm,Sm = −Sm, (c). R Q = S ,Q S = R ,S R = Q , M. Wang is with the Department of Mathematics, Qingdao University of m m m m m m m m m T T T Science & Technology, Shandong 266061, China, e-mail: ([email protected], (d). QmRm = Sm,SmQm = Rm,RmSm = Qm, [email protected]). T R R T R (e). QmA Qn = QmA Qn = A , This work is supported by the National Natural Science Foundation of T R R T R RmA Rn = RmA Rn = A , China(Grant No:11001144). T R R T R SmA Sn = SmA Sn = A International Scholarly and Scientific Research & Innovation 4(1) 2010 116 scholar.waset.org/1307-6892/4376 World Academy of Science, Engineering and Technology International Journal of Mathematical and Computational Sciences Vol:4, No:1, 2010 (f). (A∗)R =(AR)T , (AT )R =(AR)T , (A†)R =(AR)†, Next, for any λ ∈ λr(A), there exists a quaternionic vector R AM AR M R x such that Ax = xλ, and hence (g). =Π1 R R Π2, ND N D R R R A x = x diag(λ, λ, λ, λ), where Π1 and Π2 are permutation matrices. T T T R R R R (h). x Qnx = x Rnx = x Snx =0 which implies A x (:, 1) = λx (:, 1), i.e., λ ∈ λ(A ). 4n for any vector x ∈ R For any λ ∈ λ(AR), there exists a real vector u such that R R R For any matrix Y ∈ R4m×4n, Y is a real A u = λu. Because of A Rnu = λRnu, A Qnu = λQnu Theorem 2.2. R representation matrix if and only if and A Snu = λSnu,wehave R T T T A (u, Qnu, Rnu, Snu)=λ(u, Qnu, Rnu, Snu) QmYQn = RmYRn = SmYSn = Y. =(u, Qnu, Rnu, Snu)diag(λ, λ, λ, λ) It follows from theorem 2.4 that (u, Qnu, Rnu, Snu) is the Proof. Necessity. It is obvious in terms of (e) in proposition real representation of some quaternionic vector. Denote this 2.1. R R R R quaternionic vector by u1, then we have A u1 = u1 λ , and Sufficiency. Let r Au1 = u1λ, i.e., λ ∈ λ (A). ˆ T T T A Y ≡ (Y + QmYQn + RmYRn + SmYSn)/4. As we know, a Hermitian quaternionic matrix is positive (semi)definite if and only if λr(A) > 0(≥ 0). Therefore, we Yˆ = Y Y Y =(Y ) Y Then . Partition as ij 4×4, where ijs are can easily obtain the following results. m × n matrices. By direct computation, we have n×n ⎛ ⎞ Corollary 2.7. Let A ∈ Q be Hermitian. Then A is ˆ ˆ ˆ ˆ R Y1 −Y2 −Y3 −Y4 positive (semi)definite if and only if A is positive defi- ⎜ ˆ ˆ ˆ ˆ ⎟ ˆ ⎜ Y2 Y1 −Y4 Y3 ⎟ nite(semidefinite). Y = Y = ⎝ ˆ ˆ ˆ ˆ ⎠ , Y3 Y4 Y1 −Y2 Let A, B ∈ Qn×n be Hermitian. Then A − B ˆ ˆ ˆ ˆ Corollary 2.8. Y4 −Y3 Y2 Y1 is positive (semi)definite if and only if AR − BR is positive (semi)definite. where Let A ∈ R4n×4n be symmetric. Then A is Yˆ =(Y + Y + Y + Y )/4, Theorem 2.9. 1 11 22 33 44 positive (semi)definite if and only if A + QT AQ + ST AS + ˆ n n n n Y2 =(Y21 − Y12 + Y43 − Y34)/4, T R ARn is positive (semi)definite. ˆ n Y3 =(Y24 + Y31 − Y13 − Y42)/4, ˆ Proof. The necessity is obvious. Next, we prove the suffi- Y4 =(Y32 + Y41 − Y23 − Y14)/4. 4n T T T T ciency. For any x ∈ R , (I,Qn ,Rn ,Sn ) is full column rank, and therefore there exists y ∈ R4n such that So Y is the real representation matrix of quaternionic matrix ⎛ ⎞ ⎛ ⎞ ˆ ˆ ˆ ˆ Y1 + Y2i + Y3j + Y4k. I x ⎜ Q ⎟ ⎜ x ⎟ The following result may be verified directly. ⎜ n ⎟ y = ⎜ ⎟ . ⎝ R ⎠ ⎝ x ⎠ Corollary 2.3. For any Y ∈ R4m×4n, n Sn x T T T Y + QmYQn + RmYRn + SmYSn T T T T Due to (0 ≤)0 <y (A⎛+ Qn AQn + Sn AS⎞ ⎛n + Rn⎞ARn)y is a real representation matrix. A I ⎜ A ⎟ ⎜ Q ⎟ By direct verification, we easily obtain the following result. T T T T ⎜ ⎟ ⎜ n ⎟ = y (I,Qn ,Rn ,Sn ) ⎝ ⎠ ⎝ ⎠ y A Rn For any V ∈ R4m×n, Theorem 2.4. A Sn =4xT Ax, xT Ax > 0(≥ 0) A>0(≥ 0). (V,QmV,RmV,SmV ) we have , i.e., is a real representation matrix. III. POSITIVE SEMIDEFINITE SOLUTIONS OF The following result follows from (a) and (f) in proposition QUATERNIONIC MATRIX EQUATION 2.1. In this section, we discuss the relation between the positive International Science Index, Mathematical and Computational Sciences Vol:4, No:1, 2010 waset.org/Publication/4376 Theorem 2.5. U ∈ Qm×m is an unitary matrix if and only if (semi)definite solutions of quaternionic matrix equation R its real representation matrix U is an orthogonal matrix. H AXA = B (6) Theorem 2.6. Let A ∈ Qn×n be a Hermitian matrix. Then λr(A) ∈ R and and those of real matrix equation R R T R λr(A)=λ(AR), A U(A ) = B , (7) m×m where λr(A) denotes the set of right eigenvalues of A. where A ∈ Qm×n,B ∈ HQ . First, we give the relation between the general solutions of Proof. First, λr(A) ∈ R. Indeed, for Ax = xλ with the unit H ¯ H H H quaternionic matrix equation vector x,wehaveλ = x Ax, λ =(x Ax) = x Ax = λ, and therefore λ ∈ R.
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