The estimation of eigenvalues of sum, difference, and tensor product of matrices over quaternion division algebra

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1 Available online at Linear Algebra and its Applications 428 (2008) The estimation of eigenvalues of sum, difference, and tensor product of matrices over quaternion division algebra Wu Junliang, Zou Limin, Chen Xiangping, Li Shengjie College of Mathematics and Physics of Chongqing University, Chongqing , PR China Received 21 June 2007; accepted 5 February 2008 Available online 20 March 2008 Submitted by Michael Neumann Abstract In this paper, the estimation problems of eigenvalues of sum, difference, and tensor product of matrices over quaternion division algebra are studied. A pair of inequalities about eigenvalues of self-conjugate quaternion matrix is established, the famous Brauer Theorem is generalized to quaternion division algebra and the form of Cassini Theorem over quaternion division algebra is also worked out successfully Elsevier Inc. All rights reserved. AMS classification: 15A21 Keywords: Eigenvalues; Brauer Theorem; Cassini Theorem; Quaternion division algebra 1. Introduction and symbols In recent years, the algebra problem over quaternion division algebra has drawn the attention of researchers of mathematics and physics. Many problems of quaternion division algebra have been studied, such as polynomial, determinant, eigenvalues, and system of quaternion matrices equations and so on. It is not easy to study quaternion algebra problems because of the noncommutative multiplication of quaternions [1,2]. However, quaternion algebra theory is getting Foundation items: Supported by National Natural Science Foundations of China (Grant Nos and ) and Chongqing University postgraduates Science and Innovation Fund (200801A1A ). Corresponding author. address: jlwu678@tom.com (J. Wu) /$ - see front matter ( 2008 Elsevier Inc. All rights reserved. doi: /j.laa

2 3024 J. Wu et al. / Linear Algebra and its Applications 428 (2008) more and more important. In many fields of applied science, such as physics, figure and pattern recognition, space telemetry and so forth [3 8], people start making use of quaternion algebra theory to resolve some actual problems. Therefore, it is necessary make a further study about quaternion algebra theory. As a field of algebra research, quaternion matrices have been studied widely. A lot of research papers are published in various journals each year and different methods are used for different purposes [9 15], and the study of quaternion matrices is still developing. In this paper, we worked some quaternion algebra problems out theoretically; we get a pair of inequalities about eigenvalues of sum and difference of self-conjugate quaternion matrices and worked the form of Brauer Theorem and Cassini Theorem out over quaternion division algebra. Throughout this paper, we use a set of notation and terminology. R denotes set of real number. C denotes set of complex. H denotes set of real quaternion. A = (a ij ) n n denotes a n n quaternion matrix. R n n and H n n denote n n matrices set over R and H, respectively. A denotes conjugated transpose of quaternion matrix A. a = a 0 + a 1 i + a 2 j + a 3 k denotes a real quaternion (where a 0,a 1,a 2,a 3 are real numbers). α and I n denote quaternion column vector and n n unit matrix over H, respectively. Re(a) denotes real part of a. a denotes conjugated quaternion of a. α denotes conjugated transpose of quaternion vector α. N(a) = a a = aa and N(α) = α α denote the norm of quaternion a and quaternion vector α, respectively. 2. Some definitions and lemmas Definition 1. Let A H n n be given. If A = A, then A is a self-conjugate matrix over quaternion field. The set of self-conjugate quaternion matrix is remarked by H (n, ). Definition 2. Let A H n n be given. If A A = AA = I n then A is a unitary matrix over quaternion field. The set of quaternion unitary matrix is remarked by H (n, u). Definition 3. Let A H n n be given. Then, n a ii is said to be trace of quaternion matrix A, That is Tr(A) = n a ii. Definition 4. Let a be a given quaternion and let ε be a positive real number. The set Ω = {z N(z a) ε is said to be a generalized spherical neighborhood with center a and radius ε. Lemma 1 [1]. Let A H n n be given. Then,Ais a self-conjugate quaternion matrix if and only if there is a unitary matrix U H (n, u) and a real diagonal matrix diag(λ 1,λ 2,...,λ n ) such that U AU = diag(λ 1,λ 2,...,λ n ), (2.1) where λ 1,λ 2,...,λ n R are eigenvalues of A. Lemma 2. Let A H (n, ) be given. Then, for any matrix U H (n, u), we have TrA = Tr(UAU ). Proof. For any matrix U H (n, u),we have u ij u ij = N 2 (u ij ) = 1 (j = 1, 2,...,n), (2.2)

3 u ij u ij = J. Wu et al. / Linear Algebra and its Applications 428 (2008) N 2 (u ij ) = 1 (i = 1, 2,...,n). (2.3) According to Definition 3 and definition of multiplication of matrices, we have Tr(A) = Tr(UAU ). (2.4) { Lemma 3. Let A H (n, ) be given. Then, for real number t Inf α H n 1 α Aα N 2 (α),a+ ti n is a positive definite matrix. Proof. If t Inf α H n 1 { α Aα N 2 (α), (2.5) then, for any quaternion vector α H 1 n,wehave α (ti n + A)α = t(α I n α) + (α Aα) = tn 2 (α) + α Aα > 0, (2.6) According to definition of positive definite quaternion matrix, (2.6) demonstrates the validity of Lemma 3. Lemma 4 [9]. For any quaternion sequences z 1,z 2,...,z n H and w 1,w 2,...,w n H, we can get generalization Schwarz inequality over quaternion field as follows: ( ) ( )( ) N 2 z k w k N 2 (z k ) N 2 (w k ). (2.7) k=1 k=1 k=1 Proof. In [9], Lemma 4 has already been proved. So, we omit it here. Now we state and prove our main results. 3. A pair of inequalities about quaternion matrices eigenvalues Theorem 1. Let A H (n, ), C H (n, ), and B = C A. Their eigenvalues are α 1 α 2 α n,β 1 β 2 β,γ 1 γ 2 γ n, where the α i s are eigenvalues of A, the β i s are eigenvalues of B and the γ i s are eigenvalues of C. If A and C are commutative, then, n β 2 i n (γ i α i ) 2. Proof. (i) Let A be a positive definite quaternion matrix. Since A H (n, ) and C H (n, ),it is easy to get B H (n, ). Then there exist unitary matrices U 1,U 2,U 3 such that U1 AU 1 = diag(α 1,α 2,...,α n ), (3.1) U2 BU 2 = diag(β 1,β 2,...,β n ), (3.2) U3 CU 3 = diag(γ 1,γ 2,...,γ n ), (3.3) where α i > 0 (i = 1, 2,...,n). Obviously, Tr(B 2 ) = Tr[U 2 diag(β 1,β 2,...,β n )U2 U 2 diag(β 1,β 2,...,β n )U2 ] = Tr[U 2 diag(β1 2,β2 2,...,β2 n )U 2 ] = βi 2. (3.4)

4 3026 J. Wu et al. / Linear Algebra and its Applications 428 (2008) Similarly Tr(A 2 ) = Meanwhile αi 2, Tr(C2 ) = Tr(B 2 ) = Tr(A 2 AC CA + C 2 ) = γ 2 i. (3.5) αi 2 + γi 2 Tr(AC) Tr(CA). (3.6) Since A and C are commutative, so, Tr(AC) = Tr(CA) and Tr(B 2 ) = Tr(A 2 AC CA + C 2 ) = αi 2 + γi 2 2Tr(AC). (3.7) In order to prove n βi 2 (γ i α i ) 2, it is sufficient to show Tr(AC) n α i γ i Since Tr(AC) = Tr[U 1 diag(α 1,α 2,...,α n )U1 U 3 diag(γ 1,γ 2,...,γ n )U3 ] (3.8) and U1 ACU 1 = diag(α 1,α 2,...,α n )U1 U 3diag(γ 1,γ 2,...,γ n )U3 U 1, (3.9) Obviously, U = U1 U 3 = (u ij ) n n is a unitary matrix, then u ij u ij = u ij u ij = N 2 (u ij ) = 1 (j = 1, 2,...,n), (3.10) N 2 (u ij ) = 1 (i = 1, 2,...,n). (3.11) Since (AC) = C A = CA and AC = CA, then, (AC) = AC where AC H (n, ). According to Lemma 2, we have Tr(AC) = Tr(U 1 ACU 1) and Tr(AC) = Tr[diag(α 1,α 2,...,α n )U diag(γ 1,γ 2,...,γ n )U ] N 2 (u 11 ) N 2 (u 12 ) N 2 (u 1n ) γ 1 = (α 1,α 2,...,α n ) N 2 (u 21 ) N 2 (u 22 ) N 2 (u 2n ) γ 2.. (3.12) N 2 (u n1 ) N 2 (u n2 ) N 2 (u nn ) γ n We let ξ 1 N 2 (u 11 ) N 2 (u 12 ) N 2 (u 1n ) γ 1 ξ 2. = N 2 (u 21 ) N 2 (u 22 ) N 2 (u 2n ) γ 2., (3.13) ξ n N 2 (u n1 ) N 2 (u n2 ) N 2 (u nn ) γ n then it is easy to get ξ i = γ i. (3.14)

5 For any k(1 k<n),wehave ξ i = = = = J. Wu et al. / Linear Algebra and its Applications 428 (2008) N 2 (u ij )γ j γ i 1 γ i γ k γ i γ k γ i. N 2 (u ij ) γ i N 2 (u ij ) + γ k j=k+1 j=k+1 N 2 (u ij ) + γ k 1 N 2 (u ij )γ j N 2 (u ij ) N 2 (u ij ) (3.15) So, according to (3.12) and (3.13) and α i > 0 (i = 1, 2,...,n),wehave Tr(AC) = α i ξ i α i γ i. (3.16) So, n βi 2 n (γ i α i ) 2. (ii) Let A be a non-positive definite quaternion matrix. According to Lemma 3, there exists a sufficiently large real number d>0, such that A + di n is a positive definite quaternion matrix. Then, according to (i), we have (β i d) 2 [γ i (α i + d)] 2, (3.17) Furthermore β i = Tr(B) = Tr(C A) = (γ i α i ). (3.18) Thus, the result holds and proof is completed. Theorem 2. Let A H (n, ), C H (n, ), and B = A + C. Their eigenvalues are α 1 α 2 α n,β 1 β 2 β,γ 1 γ 2 γ n, where the α i s are eigenvalues of A, the β i s are eigenvalues of B and the γ i s are eigenvalues of C. If A and C are commutative, then n β 2 i n (γ i + α i ) 2 holds. Proof. In the same way above, we can finish the proof of Theorem 2. So, we omit it here. By the discussion above, we get the relationships between eigenvalues of sum and difference of two quaternion matrices. In the next section, we discuss the distribution and estimation problems about eigenvalues of quaternion matrices, some important theorems which hold on complex field will be generalized to quaternion field.

6 3028 J. Wu et al. / Linear Algebra and its Applications 428 (2008) Generalized Gerschgorin Theorem, Brauer Theorem, and Cassini Theorem over quaternion division algebra Recently, the form of the famous Gerschgorin Theorem was studied over the quaternion field [9], it solved some distribution and estimation problems of eigenvalues of quaternion matrices. We remark it here that Brauer Theorem and Cassini Theorem about complex matrices were studied in literature [17,18]. They solve the problems of the distribution and estimation of eigenvalues of tensor product of complex matrices. But there is no literature to discuss the form of Brauer Theorem over quaternion field. In this section, for the sake of applications, we first show the generalized Gerschgorin Theorem [9], then generalize the Brauer Theorem from complex field to the quaternion field. Finally, the form of Cassini Theorem over the quaternion field is worked out also. Theorem 3 [9] (The generalized Gerschgorin Theorem). Let A = (a ij ) H n n. Then all eigenvalues of A are located in the union of n generalized spherical neighborhoods {z H N(z a ii ) n 1R i, i = 1, 2...,n, (4.1) n where R i = N 2 (a ij ). We can find the proof of Theorem 3 in [9]. It can be seen that all the eigenvalues of A must lie in the set of n generalized spherical neighborhoods with central a ii and radius n 1R i. Theorem 4 (The generalized Brauer Theorem). Let A = (a ij ) H n n. Then, all eigenvalues of A are located in the union of n(n 1) 2 generalized oval neighborhoods {z H N(z a ii )N(z a jj ) R i R j, i,j = 1, 2,...n, i /= j, (4.2) ı, where R i = n N(a ij ). Proof. If λ is an eigenvalue of A, then AX = λx, (4.3) where 0 /= X = (X 1,X 2,...,X n ) T H n 1 is an eigenvector of A corresponding to λ.ifwelet N(X p ) = max N(X i ), i = 1, 2,...,n, then X p /= 0. Obviously, if the other entries of quaternion vector X are zero, then the result holds naturally. If there are at least two entries of vector X are nonzero, then we let N(X p ) = max N(X i ) and N(X q ) = max N(X i ), i /= p. So, we have X p /= 0,X q /= 0 and N(X p ) N(X q ) N(X i ). Since AX = λx, then X p (λ a pp ) = a pj X j. (4.4) j/=p

7 J. Wu et al. / Linear Algebra and its Applications 428 (2008) This means N(X p )N(λ a pp ) = N a pj X j (4.5) and Therefore N(X p )N(λ a pp ) j/=p N(a pj )N(X j ) j/=p N(a pj )N(X q ) = R i N(X q ). N(λ a pp ) N(X q) N(X p ) R P. (4.6) Meanwhile, we also have X q (λ a qq ) = a qj X j. (4.7) Then j/=q N(λ a qq ) N(X p) N(X q ) R q. (4.8) It is easy to get N(λ a pp )N(λ a qq ) N(X q) N(X p ) R N(X p ) p N(X q ) R q = R p R q, (4.9) that is N(λ a ii )N(λ a jj ) R i R j, i,j = 1, 2,...n, i /= j, (4.10) where R i = n N(a ij ). So, the proof is completed. j/=p According to Theorem 4, the follow Corollary is easy to be proved: Corollary 1. Let A = (a ij ) H n n and N(a ii )N(a jj )>R i R j,i,j = 1, 2,...,n,i /= j.then, all the eigenvalues of A are nonzero. Theorem 5. Let A = (a ij ) H n n and let λ be eigenvalue of A. Then, conjugated quaternion λ of eigenvalue λ is located in the union of n(n 1) 2 generalized oval neighborhoods {z H N(z a ii )N(z a jj ) C i C j, i,j = 1, 2,...,n, i /= j, (4.11) i, where C i = n N(a ji ). Proof. According to the proof method of Theorem 4, we can prove Theorem 5, so we leave it to the readers.

8 3030 J. Wu et al. / Linear Algebra and its Applications 428 (2008) Theorem 6. Let A = (a ij ) H n n be a central closed matrix [10]. Then all the eigenvalues of A are located in the union of n(n 1) 2 generalized oval neighborhoods {z H N(z a ii )N(z a jj ) C i C j, i,j = 1, 2,...,n, i /= j, (4.12) i, where C i = n N(a ji ). Proof. Since A = (a ij ) H n n be a central closed matrix, if λ be an eigenvalue of A, then λ R. According to Theorem 5, we have Theorem 6. Theorem 7. Let A = (a ij ) H n n be a central closed quaternion matrix. Then, for any real number α(0 α 1), all eigenvalues of A are located in the union of n(n 1) 2 generalization oval neighborhoods {z H N(z a ii )N(z a jj ) Ri α Rα j C1 α i Cj 1 α, i, i, j = 1, 2,...,n, i /= j, (4.13) where R i = n N(a ij ), C i = n N(a ji ). Proof. Let λ is an eigenvalue of A. Then, according to Theorems 4 and 6, we have {z H N(z a ii )N(z a jj ) R i R j, (4.14) i, {z H N(z a ii )N(z a jj ) C i C j. (4.15) i, So, for any real number α(0 α 1),wehave N α (λ a ii )N α (λ a jj ) Ri α Rα j (i, j = 1, 2,...,n, i /= j), (4.16) N 1 α (λ a ii )N 1 α (λ a jj ) Ri 1 α Rj 1 α (i, j = 1, 2,...,n, i /= j). (4.17) Hence N(z a ii )N(z a jj ) Ri α Rα j C1 α i Cj 1 α. (4.18) Thus, eigenvalues of A are located in the union of n(n 1) 2 generalized oval neighborhoods. That is λ {z H N(z a ii )N(z a jj ) Ri α Rα j C1 α i Cj 1 α, i, i, j = 1, 2,...,n, i /= j. (4.19) where R i = n N(a ij ), C i = n N(a ji ).

9 J. Wu et al. / Linear Algebra and its Applications 428 (2008) Literature [17,18] discussed the Cassini Theorem about the eigenvalues of complex matrix. Based on Theorems 3 and 4, we can get the form of Cassini Theorem over the quaternion field. Before stating Theorems 8 and 9, we denote the set of eigenvalues of A by λ(a). Theorem 8 (The generalized Cassini Theorem I). Let A = (a ij ) H n n be a central closed matrix and let B = (b ij ) R m m be a real symmetric matrix. Then the eigenvalues λ(a B) of tensor product A B belong to the set {z H N(z a ii ) n 1R i m {z H N(z b jj ) n 1R j. Proof. Let λ(a) ={λ 1,λ 2,...,λ m and λ(b) ={μ 1,μ 2,...,μ n be the set of eigenvalues of A and B, respectively. Then, according to Theorem 3, we have λ i {z H N(z a ii ) n 1R i, where R i = N 2 (a ij ), i = 1, 2,...,m, (4.20) μ j Meanwhile therefore m {z H N(z b jj ) n 1R j, where R j = N 2 (a ij ), j = 1, 2,...,n. (4.21) λ(a B) ={λ i μ j i = 1, 2,...,n,j = 1, 2,...,m, (4.22) λ(a B) {z H N(z a ii ) n 1R i m {z H N(z b jj ) n 1R j, (4.23) where n { z H N(z aii ) n 1R i m { z H N(z bjj ) n 1R j is the set of the product of quaternions which are located in the generalized spherical neighborhoods with the centers a ii and b jj, respectively. Theorem 9 (The generalized Cassini Theorem II). Let A = (a ij ) H n n be central closed matrix and let B = (b ij ) R m m be real symmetric matrix. Then the eigenvalues λ(a B) of tensor product A B belong to the set

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