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 ∈ 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 every 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 products 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 ∈ 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 ∈ 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 ∈ Mn(C) is positive semidefinite if σ(A) ⊆ [0, ∞) and positive definite if σ(A) ⊆ (0, ∞). Every matrix A ∈ Mn(C) has a polar decomposition A = UP , where U ∈ Mn(C) is unitary and P ∈ Mn(C) is positive semidefinite [18, Theorem 7.3.1]. Thus, every matrix is a product of two normal matrices. Another important factorization into a product of normal matrices is the singular value ∗ decomposition (SVD) A = UDV , where U, V ∈ Mn(C) are unitary matrices and D ∈ 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 nonnegative 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 ∈ 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 ∈ 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 ∈ 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 ω(x, y) = xT Jy for all x, y ∈ C2n; that is, if A ∈ Sp(2n, C), then ω(Ax, Ay) = ω(x, y). Deﬁne AJ := J −1AT J, the J-adjoint of A. From ω(Ax, Ay) = ω(x, y), we get AJ = A−1 if and only if A ∈ 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 ∈ 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 ∈ 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 ∈ 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 ∈ 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 ∈ Sp(2n, C). By [3, Theorem 32], A = A1A2A3A4, where the matrices A1,A2,A3,A4 ∈ 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 every 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 symplectic 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 symplectic 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 ∈ 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 ∈ 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 ∈ Sp(2n, C) T ∗ −1 if and only if A ,A , A, −A, A ∈ Sp(2n, C) as well. Let A ∈ M2n(C) and B ∈ 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 = ∈ 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 ∈ Sp(2m + 2n, C) if and only if A ∈ Sp(2n, C) and B ∈ 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 ∈ Sp(2n, C), then cA ∈ Sp(2n, C) if and only if c = ±1. Hence, if A ∈ 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 ∈ 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 ∈ 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). Using [20, Corollary 22], we get the following symplectic analog of the Jordan Canonical Form (see also [3, Lemma 5]).

Lemma 1. Every symplectic matrix is symplectically similar to an expanding sum of symplectic Jordan matrices of the form

−T 1. Jk(λ) ⊕ Jk(λ) , where λ ∈ C\{−1, 0, 1} and k ∈ N,

−T 2. J2k−1(λ) ⊕ J2k−1(λ) , where λ ∈ {−1, 1} and k ∈ N, and

3. ±F2k, where k ∈ N.

In particular, Lemma 1 implies that the number of Jordan blocks of a symplectic matrix corresponding to 1 or −1 of odd size must be even. We also state a symplectic analog of the spectral decomposition for normal matrices [4, Theorem 31].

Lemma 2. Let A ∈ Sp(2n, C) be normal. Then there exists a unitary U ∈ ∗ n h λj 0 i Sp(2n, ) such that UAU = −1 . C j=1 0 λj

∗ Two matrices A, B ∈ Mn(C) are said to be congruent if there exists a ∗ nonsingular X ∈ Mn(C) such that A = XBX . Note that (skew-)Hermicity is preserved by ∗congruence. Observe that if A = BC, then XAX−1 = (XBX∗)(X−∗CX−1). Thus, a similarity transformation preserves products of two matrices where each factor is either Hermitian or skew-Hermitian.

5 3 Decompositions into symplectic normal matrices

3.1 Symplectic Coninvolutions

J A matrix A ∈ M2n(C) is said to be skew-Hamiltonian if A = A, and A is J J J said to be Hamiltonian if A = −A. For any A ∈ M2n(C), AA and A A are B skew-Hamiltonian. If B ∈ M2n(C) is Hamiltonian, then A = e ∈ Sp(2n, C). The circular polar decomposition of a matrix A ∈ Mn(C) is a decomposition A = RE, where R ∈ Mn(R) and E ∈ Mn(C) such that E is a coninvolution [17, Section 7.5]. In this section, we show that if A ∈ Sp(2n, C), then A has a circular polar decomposition wherein the factors are in Sp(2n, C) as well. First, we show the following lemma.

Lemma 3. Let A ∈ Sp(2n, C) be a coninvolution. Then there exists a coninvolution R ∈ Sp(2n, C) such that R2 = A. That is, A has a square root that is both symplectic and coninvolutory.

Proof. Since A is a coninvolution, by [19, Corollary 6.4.22], we can write iS A = e for some S ∈ M2n(R), where S = −i log A. Now, SJ = J −1(−i log A)T J = −i log J −1AT J = −i log A−1 = i log A = −S. i That is, S is Hamiltonian. It follows that S is also Hamiltonian. Thus, 2 i S 2 the matrix R = e 2 is both symplectic and coninvolutory. Moreover, R = A. The circular polar decomposition with symplectic factors follows imme- diately from the previous lemma.

Theorem 2. Let A ∈ Sp(2n, C) be given. Then there exists R,E ∈ Sp(2n, C) such that R is real and E is a coninvolution.

Proof. Let A ∈ Sp(2n, C). Then A−1A ∈ Sp(2n, C) is a coninvolution, and so by Lemma 3, there exists a coninvolution E ∈ Sp(2n, C) such that E2 = A−1A. Set R = AE−1. Then A−1A = ER−1RE = E2, which gives us R = R. Thus A = RE, where R ∈ Sp(2n, C) is real and E ∈ Sp(2n, C) is a coninvolution.

6 Corollary 2. Let A ∈ Sp(2n, C) be given. If A is similar to a symplectic (skew-)coninvolution, then A is similar to a symplectic (skew-)coninvolution via a symplectic coninvolution.

Proof. Suppose A, B, X ∈ Sp(2n, C) satisfy A = X−1BX where B is a (skew-)coninvolution. By Theorem 2, X = RE for some R ∈ Sp(2n, R) and coninvolution E ∈ Sp(2n, C). Thus, A = E−1(R−1BR)E. Since real symplectic similarity preserves symplectic (skew-)coninvolutions, we get the desired result.

−1 Every coninvolution A ∈ Mn(C) can be expressed as A = X X for some X ∈ GLn(C) [18, Lemma 4.6.9]. We show that if A ∈ Sp(2n, C), we can choose X ∈ Sp(2n, C).

Lemma 4. Let A ∈ Sp(2n, C) be a coninvolution. Then A = X−1X for some X ∈ Sp(2n, C).

Proof. Suppose A ∈ Sp(2n, C) is a coninvolution. Set A−J := (AJ )−1. Note that A−J = XJ X−J . From A = A−J , we get XXJ = XXJ . That is, XXJ is real. Since XXJ is skew-Hamiltonian, by [13, Theorem 2], there exists an 2 J −1 R ∈ M2n(R) such that R is Hamiltonian and R = XX . Set Q = iR X. Then QJ = −iXJ R−1. Observe that

QJ Q = −i2XJ R−2X = XJ (XXJ )−1X = I.

That is, Q ∈ Sp(2n, C). Write X = −iRQ. Let P = iIn ⊕ −iIn ∈ Sp(2n, C). Then P −1P = −I and

A = X−1X = iQ−1R−1(−iRQ) = −Q−1Q = (PQ)−1(PQ) as desired.

3.2 Symplectic unitary and symplectic skew-Hermitian In this section, we give a “skew” analog of the polar decomposition for symplectic matrices. We start with the following lemma.

Lemma 5. Let A ∈ Sp(2n, C) be Hermitian. Then there exists a skew- Hermitian R ∈ Sp(2n, C) such that R2 = −A, σ(R) ⊆ iR+, and RA = AR.

7 Proof. Let A ∈ Sp(2n, C) be Hermitian. By Lemma 2, there exists a uni- ∗ n h λj 0 i tary U ∈ Sp(2n, ) such that B = UAU = −1 , where, since C j=1 0 λj p A is Hermitian, each λj is a positive real number. Let µj = i λj for all n h µj 0 i j ∈ {1, . . . , n} and take C = −1 . Then C ∈ Sp(2n, ) is skew- j=1 0 µj C Hermitian, C2 = −B, and CB = BC. The desired matrix is R = U ∗CU.

Let A ∈ Sp(2n, C). Then A∗A ∈ Sp(2n, C) is Hermitian. By the previous lemma, there exists a skew-Hermitian R ∈ Sp(2n, C) such that R2 = −A∗A, σ(R) ⊆ iR+, and R(A∗A) = (A∗A)R. Set Q = AR−1 ∈ Sp(2n, C). Then QQ∗ = (AR−1)(−R−1A∗) = A(−R2)−1A∗ = A(A∗A)−1A∗ = I, so Q is unitary and A = QR.

Theorem 3. Let A ∈ Sp(2n, C). Then A = QR for some unitary Q ∈ Sp(2n, C) and skew-Hermitian R ∈ Sp(2n, C). Moreover, we can choose R such that σ(R) ⊆ iR+.

3.3 Symplectic Hermitian and symplectic skew-Hermitian

Let A ∈ Mn(C) be given. If A = HS , where H is symplectic Hermitian and S is symplectic skew-Hermitian, then A = (HSH∗)H−1, where HSH∗ is symplectic skew-Hermitian and H−1 is symplectic Hermitian. Thus, we only determine when is A = HS for some symplectic Hermitian H and symplectic skew-Hermitian S.

Theorem 4. Let A ∈ Sp(2n, C) be given. The following are equivalent.

1. A = HS, where H ∈ Sp(2n, C) is Hermitian and S ∈ Sp(2n, C) is skew-Hermitian.

2. A = SC, where S ∈ Sp(2n, C) is a skew-coninvolution and C ∈ Sp(2n, C) is a coninvolution. 3. A is similar to −A−1 via a symplectic coninvolution.

4. A is similar to −A−1 via a symplectic skew-coninvolution.

5. A is similar to a symplectic skew-coninvolution.

8 6. A is similar to a symplectic skew-coninvolution via a symplectic coninvolution.

Proof. Let A ∈ Sp(2n, C). The equivalence of (1) and (2) is clear from Proposition 1 while the equivalence of (5) and (6) is Corollary 2 of Section 3.1. The remaining statements are symplectic analogs of [1, Theorem 18]. We mimic the applicable arguments. (2) implies (3) and (4): Suppose A = SC for some skew-coninvolution S ∈ Sp(2n, C) and coninvolution C ∈ Sp(2n, C). Then −A−1 = CS = CSCC−1 = CAC−1, and −A−1 = CS = S−1SCS = S−1AS. Thus, (2) implies both (3) and (4). (3) implies (2): Suppose A = −CA−1C−1 for some coninvolution C ∈ Sp(2n, C). Then −CA−1 = AC. Now,

−CA−1 = −C−1A−1 = −(AC)−1 = −(−CA−1)−1.

Hence, −CA−1 is a skew-coninvolution. (4) implies (2): Suppose A = −S−1A−1S for some skew-coninvolution S ∈ Sp(2n, C). Then −A−1S = SA. It follows that

−A−1S = A−1S−1 = (SA)−1 = (−A−1S)−1.

Hence, −A−1S is a coninvolution. (3) implies (5): Suppose −A−1 = CAC−1 for some coninvolution C ∈ Sp(2n, C). By Lemma 4, C = X−1X for some X ∈ Sp(2n, C). Thus, −A−1 = (X−1X)A(X−1X)−1 = X−1(XAX−1)X.

It follows that XAX−1 = −(XAX−1)−1 is a skew-coninvolution. (5) implies (2): Suppose A = XSX−1, where X,S ∈ Sp(2n, C) and S is a skew-coninvolution. Then A = (XSX−1)XX−1, where XSX−1 ∈ Sp(2n, C) is a skew-coninvolution and XX−1 ∈ Sp(2n, C) is a coninvolution. There is no algorithmic way of determining the conditions given in The- orem 4. The following corollary states that A satisﬁes one, and hence all, of the conditions in Theorem 4 if and only if A is similar to −A−1.

9 Suppose that A ∈ Sp(2n, C) is similar to a symplectic skew-coninvolutory −1 matrix. Then A is similar to −A . Thus, the number of Jordan blocks of A corresponding to an eigenvalue λ is the number of Jordan blocks of −1 A corresponding to −λ . Moreover, the special structure of the Jordan canonical form of a symplectic matrix implies that A is similar to the direct sum of blocks of the form:

−1 −1 1. Jk(λ) ⊕ Jk(−λ) ⊕ Jk(λ ) ⊕ Jk(−λ ) for λ 6= 0, and

2. J2k(1) ⊕ J2k(−1). For each λ 6= 0, deﬁne the symplectic matrix

−1 −T T Hk(λ) := Jk(λ) ⊕ −Jk(λ) ⊕ Jk(λ) ⊕ Jk(λ) .

−1 Suppose that A is similar to −A , so that A has a Jordan structure men- tioned above. Then A is similar to a symplectic matrix B which is the expanding sum of the blocks of the form:

1. Hk(λ) for λ 6= 0, and

−1 2. Gk := F2k −F2k . 0 Ik Deﬁne the coninvolutory matrices Xk := and Lk := Xk ⊕ Xk ∈ Ik 0 −1 Sp(2k, C). Observe that each Hk(λ) and Gk is similar to −Hk(λ) and −1 s −Gk , respectively, via Lk. Since i=1Lki is a symplectic coninvolution, B is −1 similar to −B via a symplectic coninvolution. By Theorem 4, B is similar to a symplectic skew-coninvolution. Therefore, A is similar to a symplectic skew-coninvolution. We summarize the discussion above in the following corollary.

Corollary 3. Let A ∈ Sp(2n, C). The following are equivalent.

1. A is similar to a symplectic skew-coninvolution.

2. A is similar to −A−1.

3. The number of Jordan blocks of A corresponding to λ of size k is equal to the number of Jordan blocks of A corresponding to −λ−1 of size k.

10 3.4 Two symplectic Hermitian matrices

We give necessary and suﬃcient conditions for A ∈ Sp(2n, C) to be a product of two symplectic Hermitian matrices.

Theorem 5. Let A ∈ Sp(2n, C) be given. The following are equivalent. 1. A is a product of two symplectic Hermitian matrices.

2. A is a product of two symplectic skew-coninvolutions.

3. A is similar to A−1 via a symplectic skew-coninvolution.

4. A is similar to A−1 via a skew-coninvolution.

5. The Jordan canonical of A is a direct sum of matrices of the form −1 Jk(λ) ⊕ Jk(λ ) for λ 6= 0. 6. A is similar to A∗ via a symplectic Hermitian matrix.

Proof. The equivalence of (1) and (2) is given by Proposition 1. We prove the equivalence of (2) and (3). If A = S1S2, where S1,S2 ∈ Sp(2n, C) are skew- −1 −1 −1 −1 coninvolutions, then A = S2S1 = S2AS2 . Conversely, if A = SA S for some skew-coninvolution S ∈ Sp(2n, C), then AS = SA−1. Note that (SA−1)−1 = AS−1 = −AS. Therefore, AS is a skew-coninvolution and A = ASS−1 is a product of two symplectic skew-coninvolutions. Now (3) implies (4) is trivial. The equivalence of (4) and (5) is Theorem 29 in [1]. We now show that (5) implies (1). Assume that the Jordan canonical −1 of A is a direct sum of matrices of the form Jk(λ) ⊕ Jk(λ ) for λ 6= 0. The preceding assumption, together with the Jordan canonical form restrictions of a symplectic matrix, give us that A is similar to the direct sum of matrices of the form:

−1 −1 1. Jk(λ) ⊕ Jk(λ) ⊕ Jk(λ ) ⊕ Jk(λ ) for λ 6∈ R, and

−1 2. Jk(λ) ⊕ Jk(λ) if λ ∈ R. Every complex matrix is similar to a symmetric matrix [18, Theorem 4.4.24]. For every λ ∈ R, let Ak(λ) be symmetric and similar to Jk(λ). Deﬁne the −1 −T −1 −∗ symplectic matrices Mk(λ) := Jk(λ) ⊕ Jk(λ) ⊕ Jk(λ) ⊕ Jk(λ ) and −1 Nk(λ) := Ak(λ) ⊕ Ak(λ) . Then A is similar to the symplectic B which is the expanding sum of matrices of the form

11 1. Mk(λ) for λ 6∈ R, and

2. Nk(λ) for λ ∈ R.

−1 Each Mk(λ) is similar to Mk(λ) via the symplectic skew-coninvolution −1 Jk ⊕ Jk, and each Nk(λ) is similar to Nk(λ) via the symplectic skew- −1 coninvolution Jk. The preceding statements imply that B is similar to B via a symplectic skew-coninvolution, and thus, B is a product of two symplectic Hermitian matrices, due to the equivalence of (1) and (2). Since A is symplectically similar to B, A is a product of two symplectic Hermitian matrices, as desired. Finally, we show the equivalence of (1) and (6). Suppose A = H1H2, ∗ −1 where H1,H2 ∈ Sp(2n, C) are Hermitian. Then A = H2H1 = H2AH1 and so (1) implies (6). Conversely, suppose A = HA∗H−1 for some Hermitian H ∈ Sp(2n, C). Then AH = HA∗. That is, HA∗ is Hermitian. Hence (6) implies (1). We use Theorem 5 to show that every symplectic unitary matrix is a product of three symplectic Hermitian matrices. Let P ∈ Sp(2n, C) be unitary. By Lemma 2, there exists a unitary U ∈ Sp(2n, C) such that UPU ∗ =: Q is an expanding sum of blocks of the form diag(eiθ, e−iθ). It then suffices to show that Q is a product of three symplectic Hermitian matrices. Moreover, since the expanding sum of symplectic Hermitian matrices is symplectic Hermitian, it suffices to show that 2 × 2 matrices of the form iθ −iθ Dθ := diag(e , e ) is a product of three symplectic Hermitian matrices. 1 Choose a > max , 1 and let b = a − 1 ≥ 0. We have that (Re(eiθ))2 √ √ √ √ a − beiθ aeiθ b D = √ √ √ √ . θ − be−iθ a b ae−iθ The first factor is symplectic Hermitian. Since the second factor is symplectic, −1 −1 √ iθ −iθ √its eigenvalues are λ and λ . Observe that c := λ + λ = a(e + e ) = a(2Re(eiθ)) is real, and hence λ and λ−1 are real. Also, c 6= ±2 since otherwise, 4 = 4aRe(eiθ)2, and so 1 = aRe(eiθ)2, which contradicts the choice of a. Thus λ2 6= ±1 and so the second factor is similar to the real matrix diag(λ, λ−1). By Theorem 5, the second factor is a product of two symplectic Hermitian matrices, and so we are done. Now if Q ∈ Sp(2n, C), then there exist a symplectic unitary U and symplectic Hermitian P such that Q = UP [2, Theorem 6]. We have shown that

12 U is a product of three symplectic Hermitian and thus Q is a product of four symplectic Hermitian matrices. Theorem 6. Every symplectic matrix is a product of four symplectic Her- mitian matrices.

Acknowledgment. The work of R.J. de la Cruz was funded by the UP System Emerging Inter-Disciplinary Research Program (OVPAA-EIDR-C06- 36).

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