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Products of Symplectic Normal Matrices Products of symplectic normal matrices Ralph John de la Cruz1 and Daryl Q. Granario1,2 1Institute of Mathematics, University of the Philippines, Diliman, Quezon City 1101, Philippines 2Department of Mathematics and Statistics, Auburn University, Auburn, AL 36849, USA December 16, 2017 Abstract A matrix A 2 M ( ) is symplectic if AT 0 In A = 0 In . 2n C −In 0 −In 0 We show that every symplectic matrix is a product of a symplectic unitary and a symplectic skew-Hermitian matrix. We show that ev- ery symplectic matrix is a product of four symplectic skew-Hermitian matrices or a product of four symplectic Hermitian matrices. We give the possible Jordan canonical forms of symplectic matrices which can be written as a product of a symplectic Hermitian and a matrix which is either symplectic Hermitian or symplectic skew-Hermitian. 1 Introduction A lot of work has been done in expressing matrices and operators as prod- ucts whose factors have \nice" properties. The set of normal matrices is an example of a collection of matrices frequently used and desired in applica- ∗ ∗ ∗ tions. A matrix A 2 Mn(C) is normal if AA = A A, where A denotes the conjugate-transpose of the matrix A. Normal matrices are well-studied; a list of ninety equivalent conditions can be found in the surveys [11, 15]. For E-mail address: [email protected],[email protected] 1 A 2 Cn×n, set σ(A) to be the collection of eigenvalues of A. Important sub- sets of normal matrices include the unitary matrices (A∗A = I), Hermitian matrices (A∗ = A), and skew-Hermitian matrices (A∗ = −A). A Hermitian matrix A 2 Mn(C) is positive semidefinite if σ(A) ⊆ [0; 1) and positive definite if σ(A) ⊆ (0; 1). Every matrix A 2 Mn(C) has a polar decomposition A = UP , where U 2 Mn(C) is unitary and P 2 Mn(C) is positive semidefinite [18, Theorem 7.3.1]. Thus, every matrix is a product of two normal matrices. Another im- portant factorization into a product of normal matrices is the singular value ∗ decomposition (SVD) A = UDV , where U; V 2 Mn(C) are unitary matrices and D 2 Mn(C) is a nonnegative diagonal matrix [18, Theorem 2.6.3]. Rad- javi [23] showed that every matrix A with real determinant is a product of at most four Hermitian matrices, and there are matrices which are not product of less. Taussky [24] showed that a matrix A is a product of two positive semidefinite matrices if and only if A is diagonalizable and has nonnega- tive eigenvalues. Wu [25] extended Taussky's study on positive semidefinite matrices and showed that every matrix with nonnegative determinant is a product of five positive semidefinite matrices. In the same paper, Wu also gave a characterization of matrices that can be expressed as a product of four positive semidefinite matrices. A recent work of Cui, Li, and Sze [10] gave a characterization of products of three positive semidefinite matrices. Ballantine [5, 6, 7, 8] showed that every matrix with positive determinant is a product of five positive definite matrices. We mention other decompositions of matrices whose factors are not nec- essarily normal matrices. An involution is a matrix A 2 Mn(C) such that A2 = I. Gustafson, Halmos, and Radjavi [16] proved that every matrix A with det A = ±1 is a product of four involutions. A matrix A 2 Mn(C) is a coninvolution if AA = AA = I., and A is a skew-coninvolution if AA = −I. Abara, Merino, and Paras [1] characterized matrices that are products of two coninvolutions, products of two skew-coninvolutions, and products of a coninvolution and a skew-coninvolution. We refer the reader to Wu's survey paper [26] for more information on the types of factorizations that has been considered throughout the years. We are interested in the following problem. Problem. Let G be a subgroup of GLn(C), the group of all nonsingular n × n complex matrices. Suppose A 2 G has a factorization A = P1P2 ··· Pk, where the Pj's are normal matrices with some nice properties. Can we choose the Pj's to be matrices in the group G? If yes, what are some necessary and 2 sufficient conditions for this factorization to hold in G? In this paper, we take G = Sp(2n; C), the complex symplectic group. Let J = 0 In . We drop the subscript if the size of the matrix is clear from n −In 0 context. The symplectic matrices are the isometries of the skew-symmetric bilinear form defined by !(x; y) = xT Jy for all x; y 2 C2n; that is, if A 2 Sp(2n; C), then !(Ax; Ay) = !(x; y). Define AJ := J −1AT J, the J-adjoint of A. From !(Ax; Ay) = !(x; y), we get AJ = A−1 if and only if A 2 Sp(2n; C). For symplectic matrices, a direct connection between Hermitian and skew-coninvolutions, and skew-Hermitian matrices and coninvolutions can be made. Proposition 1. Let A 2 Sp(2n; C) be given. Then 1. A is skew-Hermitian if and only if AJ and JA are coninvolutions. 2. A is Hermitian if and only if AJ and JA are skew-coninvolutions. Hence, A = QR is a product of two symplectic skew-Hermitian matrices if and only if A = (QJ)(J −1R) = (QJ)(−JR) is a product of two symplectic coninvolutions. By [3, Theorem 31], A 2 Sp(2n; C) is a product of two (symplectic) coninvolutions if and only if A is similar to its conjugate A. Thus, we have the following theorem. Theorem 1. Let A 2 Sp(2n; C) be given. Then A is a product of two symplectic skew-Hermitian matrices if and only if A is similar to A. That is, for each nonreal eigenvalue λ 2 σ(A), if A has m Jordan blocks corresponding to λ of size k, then A has m Jordan blocks correponding to λ of size k. Let A 2 Sp(2n; C). By [3, Theorem 32], A = A1A2A3A4, where the matrices A1;A2;A3;A4 2 Sp(2n; C) are coninvolutions. It follows that we can −1 −1 write A = (A1J )(JA2)(A3J )(JA4), where all the factors are symplectic skew-Hermitian. Corollary 1. Every symplectic matrix is a product of four symplectic skew- Hermitian matrices. In Section 2, we give some preliminary results. In Section 3, we prove our main results and some further findings. In particular, we show that ev- ery symplectic matrix may be written as a product of a symplectic unitary 3 and a symplectic skew-Hermitian matrix (Theorem 3). We show that a sym- plectic matrix A is a product of a symplectic Hermitian and a symplectic skew-Hermitian matrix if and only if A is similar to −A−1 (Theorem 4 and Corollary 3). We give restrictions to the Jordan canonical form of a sym- plectic matrix A for A to be a product of two symplectic Hermitian matrices (Theorem 5). We use the preceding result to show that every matrix is a product of four symplectic Hermitian matrices (Theorem 6). 2 Preliminaries We start by recalling some basic facts regarding symplectic matrices. Let A = A11 A12 2 Sp(2n; ), partitioned into a 2 × 2 block matrix with n × n A21 A22 C J blocks Aij. From AA = I, we get T T 1. A11A22 − A12A21 = I, and T T 2. both A11A12 and A21A22 are symmetric. −T Consequently, for any X 2 GLn(C), the matrix X ⊕ X is symplectic, where A ⊕ B denotes the usual direct sum of the square matrices A and B, and A−T := (A−1)T for a nonsingular matrix A. Note that A 2 Sp(2n; C) T ∗ −1 if and only if A ;A ; A; −A; A 2 Sp(2n; C) as well. Let A 2 M2n(C) and B 2 M2m(C). Partition the matrices as 2 × 2 block matrices with n × n blocks A = [Aij] and m × m blocks B = [Bij], respectively. The expanding sum of A and B [22, 9] is the matrix A11 ⊕ B11 A12 ⊕ B12 A B = 2 M2m+2n(C): A21 ⊕ B21 A22 ⊕ B22 The matrix A B is permutation similar to A ⊕ B. The expanding sum preserves symplecticity, that is, A B 2 Sp(2m + 2n; C) if and only if A 2 Sp(2n; C) and B 2 Sp(2m; C). The expanding sum of Hermitian, skew-Hermitian, or unitary matrices is also, respectively, Hermitian, skew- Hermitian, or unitary. For even sized matrices A1;A2;:::;An, we use the notation m i=1Ai := A1 A2 ··· An: If Ai and Bi are of the same even size for i = 1; 2; : : : ; m, then m m m (i=1Ai)(i=1Bi) = i=1AiBi: 4 If A 2 Sp(2n; C), then cA 2 Sp(2n; C) if and only if c = ±1. Hence, if A 2 Sp(2n; C) is Hermitian, then iA is skew-Hermitian but not symplectic. Thus, the study of symplectic Hermitian matrices is not equivalent to the study of symplectic skew-Hermitian matrices. If A; B 2 Sp(2n; C) are similar, then the matrix of similarity can be taken to be symplectic [20, Corollary 22]. Denote by Jk(λ) the k-by-k upper triangular Jordan block corresponding to λ. For k 2 N, define Jk(1) Uk F2k := −T ; 0 Jk(1) −T where Uk := (0k−1;k−1 ⊕ [1])Jk(1) . The matrix F2k is a symplectic matrix that is similar to J2k(1).
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