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 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 [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 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 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 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

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(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 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 if and only if (semi)definite solutions of quaternionic matrix equation R its real representation matrix U is an . 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. AXC = E (8)

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and those of real matrix equation IV. CONCLUSION R R T R A U(C ) = E , (9) In the paper, we only take a simple but common equation (6) as an example. Our idea is applied to more complicated A ∈ Qm×n,C ∈ Qp×q,E ∈ Qm×q where . The following linear quaternionic matrix equations. result is a special case of the corresponding result of [7].

Lemma 3.1.Quaternionic matrix equation (8) has a solution REFERENCES n×p X ∈ Q if and only if real matrix equation (9) has a Foundations of 4n×4p [1] D. Finkelstein , J. Jauch , S. Schiminovich and D. Speiser, solution U ∈ Q , in which case, quaternion quantum mechanics, J. Math. Phys., vol. 3, 1962, pp.207-231. [2] S. Adler, Quaternionic quantum field theory Commun. Math. Phys., vol. 1 T X = (In,iIn,jIn,kIn)(U + Qn UQp 104, 1986, pp. 611-623. 16 ⎛ ⎞ [3] S. Adler, Quaternionic Quantum Mechanics and Quantum Fields,New Ip York: Oxford University Press, 1995. ⎜ −iI ⎟ [4] J. Jiang, An algorithm for quaternionic linear equations in quaternionic T T ⎜ p ⎟ quantum theory +Rn URp + Sn QSp) ⎝ ⎠ , J. Math. Phys., vol. 45, 2004, pp.4218-4228. −jIp [5] J. Jiang, Cramer ruler for quaternionic linear equations in quaternionic −kIp quantum theory, Rep. Math. Phys., vol. 57, 2006, pp. 463-467. [6] J. Jiang, Algebraic algorithms for least squares problem in quaternionic is a quaternionic matrix solution of (8). Furthermore, if (9) quantum theory, Comput. Phys. Commun., vol. 176, 2007, pp. 481-485. has an unique solution, then (8) has also an unique solution. [7] J. Jiang, Real representiations of quaternion matrices and quaternion matrix equations, Acta Mathematica Scientia, vol. 26A, 2006, pp. 578- m×m Theorem 3.2. Given A ∈ Qm×n,B ∈ HQ . Then 584. 1. (6) has a positive (semi)definite solution if and only if [8] M. Wang, M. Wei and Y. Feng, An iterative algorithm for least squares problem in quaternionic quantum theory, Comput. Phys. Commun., vol. real matrix equation (7) has a positive (semi)definite solution. 179, 2008, pp. 203-207. 2. When (6) has a positive (semi)definite solution, the general expression of this solution is

1 T X = (In,iIn,jIn,kIn)(U + Qn UQp 16 ⎛ ⎞ Ip ⎜ ⎟ T T ⎜ −iIp ⎟ +Rn URp + Sn QSp) ⎝ ⎠ , −jIp −kIp where U is a positive (semi)definite solution of (7). Fur- thermore, if U is the maximal(minimal) solution of (7), then the corresponding solution X is also the maximal(minimal) solution of (6). Proof. If (7) has a positive (semi)definite solution U, then from Theorem 2.9 and Theorem 3.1, we know that 1 Uˆ ≡ (U + QT UQ+ ST US + RT UR) 4 is also a positive (semi)definite solution of (7). Let Uˆ be the real representation matrix of quaternionic matrix X. It follows from Corollary 2.7 that X is a positive (semi)definite solution of (6). If U1 and U2 are positive (semi)definite solutions of (7) satisfying U1 ≥ U2, then 1 (U + QT U Q + ST U S + RT U R) 4 1 1 1 1

1 T T T ≥ (U2 + Q U2Q + S U2S + R U2R), International Science Index, Mathematical and Computational Sciences Vol:4, No:1, 2010 waset.org/Publication/4376 4 and both are positive (semi)definite solutions of (7). Let them be the real representation matrices of quaternionic matrces X1 and X2, respectively. Then from Corollary 2.8, we have X1 and X2 are positive (semi)definite solutions of (6) satisfying X1 ≥ X2. Theorem 3.2 establishs the relation between positive (semi)definite solutions of quaternionic matrix equations (6) and those of corresponding real matrix equations (7). For the latter, there have been many good theoretical results and numerical methods, which may be applied to the former.

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