Unit Triangular Factorization of the Matrix Symplectic Group

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• Block LDU parametrization. Before describing the block LDU parametrization, we ﬁrst show the deﬁnition of symplectic interchange.

Deﬁnition 2. Let 1 ≤ j ≤ d. The symplectic interchange matrix Πj is the 2d-by-2d matrix obtained by interchanging the columns j and j + d of the 2d-by-2d identity matrix and multiplying the j-th column of the resulting matrix by −1. The symplectic interchange matrix Πj is the 2d-by-2d matrix obtained by interchanging the columns j and j + d of the 2d-by-2d identity matrix and multiplying the j + d-th column of the resulting matrix by −1. Notice T thate Πj = Πj.

With the abovee deﬁnition, there is the block LDU parametrization.

Theorem 1. The set of 2d-by-2d symplectic matrices is

I 0 G 0 I E d×d SP = Q −T G ∈ R nonsingular, ( C I 0 G 0 I

C = CT , E = ET , Q a product of symplectic interchanges . )

The detailed information of the block LDU parametrization can be found in [12].

• QR-like factorization.

Theorem 2. The set of 2d-by-2d symplectic matrices is

× R Z R ∈ Rd d upper triangular SP = Q −T . ( 0 R Q symplectic orthogonal )

The detailed information of the QR-like factorization can be found in [9].

• Polar factorization.

2 Theorem 3. The set of 2d-by-2d symplectic matrices is

P = P T symplectic positive definite SP = QP . ( Q symplectic orthogonal )

The detailed information of the polar factorization can be found in [21].

• SVD-like factorization.

Theorem 4. The set of 2d-by-2d symplectic matrices is

U, V symplectic orthogonal Ω 0 T SP = U −1 V Ω= diag(ω1, ···,ωd)  . 0 Ω  ω1 ≥···≥ ωd ≥ 1 

  The detailed information of the SVD-like factorization can be found in[36].

• Transvections factorization. In text books on classical groups like [4, 33, 35] this result is mostly stated as a corollary of another basic fact, namely that the symplectic group is generated by so-called symplectic transvections.

Deﬁnition 3. For 0 =6 u ∈ R2d and 0 =6 β ∈ R, the matrix

× G = I + βuuT J ∈ R2d 2d

is symplectic, and G is called a symplectic transvection.

Any symplectic matrix can be factored as the product of many symplectic transvections.

Theorem 5. The set of 2d-by-2d symplectic matrices is

Gi a symplectic transvection SP = G1G2 ··· Gm . ( m ≤ 4d )

Although this factorization can freely parameterize the symplectic matrices, in numerical practice, the computed product of lots of symplectic matrices may be far from being symplectic when d is large. There are some works parameterizing the symplectic matrices as the ﬁnite product of certain elementary units [24, 30], however, which take eﬀect on only special structured symplectic matrices. The detailed information of the transvections factorization can be found in [16, 27].

In this work, we will show a more elementary factorization of the symplectic matrices. Consid- ering I S I 0 , , (1) 0 I S I

3 where S is symmetric, we will factor the symplectic matrices as the products of at most 9 unit triangular symplectic matrices shown above. In addition, we call the symplectic matrices like

P 0 (2) 0 P −T as diagonal symplectic matrices. These two types of symplectic matrices are concerned as the basic units in our factorizations. Since the symplectic matrix with nonsingular left upper block is easy to deal with, in classical factorizations, researchers have to exploit some intractable factors such as symplectic interchanges and permutation matrices to make a general symplectic matrix equipped with nonsingular left upper block. Different from the conventional methods, we find a family of unit triangular symplectic matrices which can transform a general symplectic matrix into one having nonsingular left upper block, it is the core of the theoretical results in this work. With this result, we are able to factor the symplectic matrix into three unit triangular symplectic matrices and one diagonal symplectic matrix, called unit ULU factorization. Furthermore, we decompose the diagonal symplectic matrix into several unit triangular symplectic matrices and subsequently obtain the final unit triangular factorization. The unit triangular factorization immediately leads to many significant properties of the matrix symplectic group, such as, (i) the determinant of symplectic matrix is one, (ii) the matrix symplectic group is path connected, (iii) all the unit triangular symplectic matrices form a set of generators of the matrix symplectic group. In addition, we will consider about some structured subsets of the matrix symplectic group such as the symplectic matrices with nonsingular left upper block, the positive definite symplectic matrices and the symplectic M-matrices, which can be factored into less than 9 unit triangular symplectic matrices due to their special structures. A remarkable characteristic of the unit triangular factorization is that it provides an unconstrained parametrization for symplectic matrices. For the general symplectic matrices, the positive definite symplectic matrices and the singular symplectic matrices, we will explore their unconstrained parametrization respectively. With the unconstrained parametrization, one may be able to apply faster and more efficient unconstrained optimization algorithms to the problems with symplectic constraints under certain circumstances, which will be discussed in detail later. It was in fact our work on optimization problem with symplectic constraints that led us to consider the unit triangular factorization of the symplectic matrices. The remaining parts of this paper are organized as follows. Some basic properties that will be used later are given in Section 2. Section 3 presents the detailed proofs of the factorizations and deduces several important inferences. In Section 4, we provide the unconstrained parametrization for the matrix symplectic group and its structured subsets. Some conclusions will be given in the last section.

2 Preliminaries

At first, we introduce the notations will be used as well as some relevant basic results for factorization of symplectic matrices. As shown in Definition 1, in this work we consider the real 2d-by-2d matrix symplectic group SP . Property 1-3 are well-known and can be easily verified according to the definition of SP . We omit the proofs and the readers are referred to [10, 14] for more details.

4 Since the features of symplectic matrices are not apparent in Deﬁnition 1, there is an equivalent condition of being symplectic. A B Property 1. If H = 1 1 ∈ R2d×2d is a symplectic matrix and A , B ∈ Rd×d, then A B i i 2 2 T T (i) A1 A2 = A2 A1,

T T (ii) B1 B2 = B2 B1,

T T (iii) A1 B2 − A2 B1 = I, A vice versa. Note that an A = 1 ∈ R2d×d satisfying the ﬁrst item is called a symmetric pair. A 2 Property 1 points out the detailed relationship among the four blocks in a symplectic matrix, it plays an important role on unpacking the features. Based on Property 1, one may check that a symplectic matrix being unit and triangular must be written in form (1). Besides the inner structure of symplectic matrix, there also hold several overall natures, for instance, SP remains closed under multiplication, transposition and inversion. Property 2. If H,G ∈ SP , then (i) HG ∈ SP ,

(ii) HT ∈ SP ,

(iii) H−1 ∈ SP . With the closure under multiplication and inversion, we can easily check that SP immediately forms a group. In addition, the second item in Property 2 indicates that the equivalent condition in Property 1 can be written in another way, as

T T A1B1 = B1A1 T T  A2B2 = B2A2 . (3)  T T  A1B2 − B1A2 = I In this work, we are aiming to seek elementary factorizations based on only (1) and (2), which are the most concise symplectic matrices. To figure out how the unit triangular matrices and the diagonal matrices are being symplectic, here shows a property as follows. Property 3. If S,P,Q,A,B,C ∈ Rd×d, then I S (i) The matrix is symplectic if and only if ST = S, 0 I I 0 (ii) The matrix is symplectic if and only if ST = S, S I P 0 (iii) The matrix is symplectic if and only if Q = P −T . 0 Q 5 The mentioned two types of matrices, i.e., the unit triangular symplectic matrices and the diagonal symplectic matrices, have satisfactory properties and clear structures. In the later sections we will think of a way to do the unconstrained parametrization of symplectic matrices and its structured subsets, for this purpose, the unit triangular symplectic matrices (1) are more preferred, d(d+1) since each one is determined by 2 free parameters while the diagonal symplectic matrices (2) require the nonsingularity on P . Although the determinant of symplectic matrix is obviously ±1, it is not easy to confirm that the determinant is always 1. However, in some special cases, we can prove it without any difficulty. For instance, when the symplectic matrix has a nonsingular left upper block, we immediately derive such a LDU factorization whose factors are simple enough to obtain the determinants.

A1 B1 2d×2d Property 4 (LDU Factorization). If H = ∈ R is a symplectic matrix and Ai, Bi ∈ A2 B2 d×d R , moreover A1 is nonsingular, then H has three unique factorizations

P1 0 I 0 I T1 H = −T 0 P1 S1 I 0 I , (4)  T −1  S1 = A1 A2, T1 = A1 B1, P1 = A1

 I 0 P2 0 I T2 H = −T S2 I 0 P2 0 I , (5)  −1 −1  S2 = A2A1 , T2 = A1 B1, P2 = A1

 I 0 I T3 P3 0 H = −T S3 I 0 I 0 P3 , (6)  −1 T  S3 = A2A1 , T3 = B1A1 , P3 = A1 where S ,S ,S , T , T , T are symmetric and P ,P ,P are nonsingular. 1 2 3 1 2 3  1 2 3

Proof. Here we show the proof of (5), and the others are the same. Assume that S2, T2,P2 are deﬁned as in (5), then

T −1 T −T T −1 −T T −1 −1 S2 =(A2A1 ) = A1 A2 A1A1 = A1 A1 A2A1 = A2A1 = S2, T −1 T −1 T −T −1 T −T −1 T2 =(A1 B1) = A1 A1B1 A1 = A1 B1A1 A1 = A1 B1 = T2, which mean S2, T2 are symmetric. Furthermore,

−T −T −1 −1 −T T −1 P2 + S2P2T2 = A1 + A2A1 A1A1 B1 = A1 (I + A1 A2A1 B1) −T T −1 = A1 (I + A2 A1A1 B1) −T T = A1 A1 B2

= B2, therefore

I 0 P2 0 I T2 P2 P2T2 A1 B1 − = − = = H. S I 0 P T 0 I S P P T + S P T A B 2 2 2 2 2 2 2 2 2 2 The uniqueness is easy to check and we omit it.

6 Property 4 decomposes the symplectic matrices with nonsingular left upper block into two unit triangular symplectic matrices and one diagonal symplectic matrices, where the diagonal one can be placed at an arbitrary position. What we expect is to derive a similar result for the general symplectic matrices. In order to complete our proofs, we also have to give the other properties for checking if some columns are a part of a symplectic matrix. The relevant properties can be found in [14, 18]. A Property 5. If A = 1 ∈ R2d×k, 1 ≤ k ≤ d, A is full-rank, and AT A = AT A , then there A 1 2 2 1 2 exists a A′ ∈ R2d×(d−k) such that A A′ is a full-rank symmetric pair. Proof. It is shown as [14, p. 147, Theorem 3.27]. Property 6. If A ∈ R2d×d is a full-rank symmetric pair, then there exists a B ∈ R2d×d such that A B ∈ SP . Proof. It is shown as [14, p. 147, Theorem 3.28]. At the end of this section, we provide a table including the main notations in this paper. See Table 1.

3 Unit triangular factorization

In this section, we will present the detailed proof of the unit triangular factorization. To describe the problem clearly, denote

I 0/Sn I 0 I S1 d×d T Ln = ··· Si ∈ R ,Si = Si, i =1, 2, ··· , n , ( Sn/0 I S2 I 0 I ) where the unit upper triangular symplectic matrices and the unit lower triangular symplectic matrices appear alternately. And it is clear that Lm ⊂Ln ⊂ SP for all integers 1 ≤ m ≤ n. Now the main theorem is given as follows.

Theorem 6 (Unit Triangular Factorization). SP = L9. Thus any symplectic matrix can be factored into no more than 9 unit triangular symplectic matrices. To prove this theorem, we first list some necessary auxiliary results. For convenience, we denote all the unimportant blocks in matrices by “⋆” throughout this paper. Theorem 7. For any symplectic matrix H ∈ R2d×2d, there exists a symmetric S ∈ Rd×d such that, the factorization I λS P ⋆ H = λ 0 I ⋆ ⋆ holds with a nonsingular Pλ for all λ =06 . Hence any symplectic matrix can be decomposed into a unit upper triangular symplectic matrix and a symplectic matrix with nonsingular left upper block. Furthermore, if needed, S can be set to O 0 S = −P r P T 0 I − d r 7 R2d×2d All the symplectic matrices involved are in the real space R2d×2d. 0 I J d . −I 0 d SP The collection of all the 2d-by-2d symplecic matrices. SP N The collection of all the 2d-by-2d symplecic matrices with nonsingular left upper block. SPP The collection of all the 2d-by-2d symmetric positive definite symplecic matrices. SP M The collection of all the 2d-by-2d symplecic M-matrices. The definition of M-matrices is shown in Definition 4. SPS The collection of all the 2d-by-2d singular symplecic matrices. The definition of singular symplecic matrices is shown in Definition 5.

I 0/Sn I 0 I S1 d×d T Ln ··· Si ∈ R ,Si = Si . ( Sn/0 I S2 I 0 I ) The unit upper triangular symplectic matrices and the unit

lower triangular symplectic matrices appear alternately. 2 T Ln {LL |L ∈Ln}. A subset of SPP . T T L1 {L |L ∈L1}. The collection of all the unit lower triangular symplectic matrices. T SG L1 ∪L1 . A set of generators of the group SP . P a The map for extracting the lower triangular parameters for symmetric matrices. P a(S)=(s11,s21,s22,s31, ··· ,sdd), d×d T where S =(sij) ∈ R , S = S.

Table 1: The main notations in this paper.

8 when the left upper block A1 of H with rank r is decomposed as

Ir 0 A1 = P Q 0 O − d r where P, Q ∈ Rd×d are nonsingular. A B Proof. Denote H = 1 1 ∈ SP where A , B ∈ Rd×d, and assume that rank(A )= r. There A B i i 1 2 2 −1 −1 Ir 0 Or 0 T exist nonsingular P, Q such that P A1Q = , and let S = −P P , λ =6 0. 0 Od−r 0 Id−r Then

−1 −T −1 −1 I −λP SP P 0 A1 B1 Q 0 0 I 0 P T A B 0 QT 2 2 Ir 0 Or 0 ⋆ I λ 0 Od−r = 0 Id−r     C1 C2 0 I ⋆  C C     3 4  I 0   r ⋆ = λC λC  3 4  ⋆ ⋆ D ⋆  = λ ⋆ ⋆ I 0 where C ∈ Rr×r,C ∈ Rr×(d−r),C ∈ R(d−r)×r,C ∈ R(d−r)×(d−r) as well as D = r , 1 2 3 4 λ λC λC 3 4 next we explain why Dλ is nonsingular. Since all the matrices involved above are symplectic, we I 0 C C know that r and 1 2 satisfy 0 O − C C d r 3 4 I 0 T C C C C T I 0 r 1 2 = 1 2 r 0 O − C C C C 0 O − d r 3 4 3 4 d r due to Property 1, which implies that C2 = 0. Furthermore,

Ir 0 Ir 0 0 O − 0 0  d r =   C1 C2 C1 0 C C  C C   3 4   3 4     is full-rank as the sub-columns of a symplectic matrix, thus C4 is also full-rank, i.e., nonsingular. I 0 Hence D = r is nonsingular. λ λC λC 3 4

9 Now we have A B P 0 I λP −1SP −T D ⋆ Q 0 H = 1 1 = λ A B 0 P −T 0 I ⋆ ⋆ 0 Q−T 2 2 P λSP −T D ⋆ Q 0 = λ 0 P −T ⋆ ⋆ 0 Q−T I λS P 0 D ⋆ Q 0 = λ 0 I 0 P −T ⋆ ⋆ 0 Q−T I λS PD Q ⋆ = λ 0 I ⋆ ⋆ I λS P ⋆ = λ 0 I ⋆ ⋆ where Pλ = PDλQ is nonsingular for all λ =6 0. Remark 1. In the most cases, we are unconcerned about the λ in Theorem 7, and only focus on the factorization I S P ⋆ H = 0 I ⋆ ⋆ where S is symmetric and P is nonsingular. It in fact yields a symplectic matrix with nonsingular left upper block which is much more easier to deal with. To prove the determinant of any symplectic matrix is one, researchers make eﬀorts to have the symplectic matrix equipped with nonsingular left upper block, however, all of their transformations require the intractable blocks like symplectic interchanges and permutation matrices. Theorem 7 is the core of the proof of Theorem 6, it transforms a symplectic matrix into one having nonsingular left upper block by one unit triangular symplectic matrix. Lemma 1 (Unit ULU Factorization). For any symplectic matrix H ∈ R2d×2d, there exist symmetric S,T,U ∈ Rd×d and a nonsingular P ∈ Rd×d such that I S P ⋆ I S I 0 I U P 0 H = = . 0 I ⋆ ⋆ 0 I T I 0 I 0 P −T Furthermore, T,U,P are uniquely determined by H and S. Proof. Theorem 7 shows that there exists the factorization I S P ⋆ H = 0 I ⋆ ⋆ where S is symmetric and P is nonsingular. Moreover, by (6),

P ⋆ P A I 0 I AP T P 0 = = , ⋆ ⋆ B C BP −1 I 0 I 0 P −T hence I S I 0 I U P 0 H = 0 I T I 0 I 0 P −T 10 where T = BP −1 and U = AP T are symmetric. In addition, when H and S are ﬁxed, we are able to obtain P,A,B,C,T,U in sequence, thus the factorization is determined by H,S. Note again that all the matrices involved above are symplectic.

Remark 2. Notice that P 0 I S I PSP T P 0 = . 0 P −T 0 I 0 I 0 P −T P 0 The diagonal symplectic matrices like can be moved to any position between a series of 0 P −T I ⋆ I 0 many and . For instance, 0 I ⋆ I P 0 I 0 I ⋆ I 0 P 0 I ⋆ = 0 P −T ⋆ I 0 I ⋆ I 0 P −T 0 I I 0 I ⋆ P 0 = , ⋆ I 0 I 0 P −T therefore we can adjust the position of diagonal symplectic matrices among the unit triangular symplectic matrices by moving, even merge them into one. What we just did is applying the LDU factorization to Theorem 7 in the proof of Lemma 1. According to Remark 2, Lemma 1 has some other forms, like I S I 0 P 0 I ⋆ H = 0 I ⋆ I 0 P −T 0 I and I S P 0 I 0 I ⋆ H = . 0 I 0 P −T ⋆ I 0 I The most important fact is the number of unit triangular matrices, rather than the position of diagonal matrix. Note that all the forms involved share the same P due to Property 4. Actually, Theorem 7 and Lemma 1 provide an algorithm for unit ULU factorization, see Algo- rithm 1. So far we have proved that any symplectic matrix can be factored into three unit triangular symplectic matrices together with one diagonal symplectic matrix, where the number three is optimal, since the product of two only yields a symplectic matrix with nonsingular left upper block. To obtain the ﬁnal unit triangular factorization, we have to consider about how to decompose the diagonal symplectic matrices next.

P 0 2d×2d d×d Lemma 2. For any symplectic matrix H = − ∈ R where P ∈ R is nonsingular, 0 P T d×d d×d there exist symmetric S1,S2, ··· ,S7 ∈ R as well as symmetric T1, T2, ··· , T7 ∈ R such that I S I 0 I 0 I S H = 7 ··· 1 0 I S I S I 0 I 6 2 I 0 I T I T I 0 = 6 ··· 2 . T I 0 I 0 I T I 7 1 11 Algorithm 1 Unit ULU factorization Input: H ∈ SP I S I 0 I U P 0 Output: S,T,U,P such that H = 0 I T I 0 I 0 P −T A ⋆ Given H := ∈ SP ⋆ ⋆ I 0 Compute the factorization A = P r Q where P, Q are nonsingular 0 O − d r O 0 Set S := −P r P T 0 I − d r A B I −S Set 1 1 := H A2 B2 0 I −1 Set T := A2A1 T Set U := B1A1 Set P := A1 return S,T,U,P

d×d Proof. At ﬁrst, there exist two symmetric matrices P1,P2 ∈ R such that P = P1 · P2, then we have P 0 P1 0 P2 0 H = = − − . 0 P −T 0 P T 0 P T 1 2 Moreover, P1 0 I −P1 I 0 I I I 0 − = − , 0 P T 0 I P 1 − I I 0 I P − I I 1 1 1 −1 P2 0 I 0 I P2 − I I 0 I P2 − I − = − , 0 P T −P 1 I 0 I I I 0 I 2 2 so that

I −P1 I 0 I I I 0 H = − − 0 I P 1 − I I 0 I P − I − P 1 I 1 1 2 I P − I I 0 I P −1 − I · 2 2 . 0 I I I 0 I T Up to now, we have obtained the expressions of Si exactly. For Ti, we just need to express H in the above way and return to H by transposition operator again.

Remark 3. As shown in the proof of Lemma 2, some of the blocks can be required to be Id for reducing the degree of freedom of factorization, such as S2,S5. Now we are able to provide the proof of Theorem 6. Proof. Let H ∈ SP . According to Lemma 1, we have

I S I 0 I U P 0 H = . 0 I T I 0 I 0 P −T 12 where S,T,U are symmetric and P is nonsingular. Lemma 2 shows that

P 0 I S I 0 I 0 I S = 7 ··· 1 , 0 P −T 0 I S I S I 0 I 6 2 so that I S I 0 I U I S I 0 I 0 I S H = 7 ··· 1 0 I T I 0 I 0 I S I S I 0 I 6 2 I S I 0 I U + S I 0 I 0 I S = 7 ··· 1 , 0 I T I 0 I S I S I 0 I 6 2 thus H has been decomposed into 9 unit triangular symplectic matrices.

Remark 4. This factorization starts with an upper triangular matrix in right-hand side. In fact, if we decompose the HT into the above 9 factors, subsequently obtain an expression of H starting with a lower triangular matrix by transposition.

Theorem 6 shows an elegant expression of symplectic matrix, and it implies several important properties. Corollary 1. The determinant of any symplectic matrix is one.

Proof. The determinant of any unit triangular symplectic matrix is one. There are many proofs for the determinant of symplectic matrix [8, 27]. Compared to the conventional proofs for the determinant of symplectic matrix, such as QR-like factorization, SVD- like factorization and transvections factorization, Theorem 6 is the most symplectic-determinant- revealing and elementary. Corollary 2. The matrix symplectic group SP is path connected. Proof. Given two symplectic matrices H,G ∈ SP , we can express them as

I S I 0 I S H = 9 ··· 1 0 I S I 0 I 2 I T I 0 I T G = 9 ··· 1 0 I T I 0 I 2 where Si, Ti are symmetric, due to Theorem 6. Deﬁne the continuous γ : [0, 1] → SP as

I (1 − t) · S + t · T I 0 I (1 − t) · S + t · T γ(t)= 9 9 ··· 1 1 , 0 I (1 − t) · S + t · T I 0 I 2 2 then we have γ(0) = H and γ(1) = G, thus SP is path connected.

T T T Corollary 3. Denote SG = L1 ∪L1 where L1 := {H |H ∈L1}, then SG is a set of generators of the group SP . Proof. It is obvious by Theorem 6.

13 Remark 5. Although Corollary 1, 2 are well-known and can be found in many literatures such as [10, 14], here we provide the alternative and concise proofs which are deduced directly from the unit triangular factorization. It is noteworthy that Corollary 3 is indeed a new feature, while the existing theory of generators of SP requires the factors of diagonal symplectic matrices (2). For example, [10, p. 49, Corollary 2.40] states that {J}∪{unit lower triangular symplectic matrices} ∪ {diagonal symplectic matrices} is a set of generators of SP . In fact, Corollary 3 indicates that {diagonal symplectic matrices} can be removed from above statement.

Up to now, we have studied several properties of the general symplectic matrices. Next we focus on the structured sets of symplectic matrices.

3.1 Symplectic matrices with nonsingular left upper block For symplectic matrices with other structures, which are naturally subsets of SP , we can exploit their special structures for reducing 9 to a smaller number. Here we turn to the desirable symplectic matrices which have nonsingular left upper blocks. Denote

A ⋆ × SP N = H = ∈ SP A ∈ Rd d nonsingular . ( ⋆ ⋆ )

We have proved that SP = L9, furthermore, it is easy to show that SP N ( L8.

Theorem 8. SP N ( L8. Thus any symplectic matrix with nonsingular left upper block can be factored into no more than 8 unit triangular symplectic matrices. Proof. With the LDU factorization and Lemma 2, we know

A ⋆ I 0 A 0 I T H = = ⋆ ⋆ S I 0 A−T 0 I I 0 I S I 0 I S + T = 7 ··· 1 , S I 0 I S I 0 I 2 I I I 0 I I so that SP N ⊂ L . Moreover, consider J = , we have J ∈ L ⊂ L and 8 0 I −I I 0 I 3 8 J∈ / SP N, hence SP N ( L8.

Until now, we have SP N ( L8 ⊂L9 = SP , however, it is not sure whether L8 = L9 or L8 ( L9. [12] provides a proof that SP N is a dense subset in the group of symplectic matrices, although it can be accomplished through general properties of algebraic manifolds. Here we present a simple proof again according to Theorem 7, and this result immediately shows that L8 is at least dense in L9 = SP . Corollary 4. SP N is dense in SP , i.e., SP N = SP . Proof. Theorem 7 shows that, for any H ∈ SP , there exists a symmetric S such that

I λS P ⋆ H = λ 0 I ⋆ ⋆ 14 holds with a nonsingular Pλ for all λ =6 0. Therefore I −λS P ⋆ H = lim H = lim λ ∈ SPN, λ→0 0 I λ→0 ⋆ ⋆ which means SP ⊂ SP N, consequently SP = SP N.

Corollary 5. L8 is dense in L9, i.e., L8 = L9. Proof. SP N is dense in SP .

3.2 Positive deﬁnite symplectic matrices Denote the collection of symmetric positive deﬁnite symplectic matrices by SPP = {H ∈ SP |H symmetric positive definite}.

Theorem 9. SPP ( L4. Thus any symmetric positive definite symplectic matrix can be factored into no more than 4 unit triangular symplectic matrices. Proof. If H is symmetric positive definite, then all principal submatrices of H are symmetric pos- H1 H2 itive definite. Denote that H = ∈ SPP , here H1 is positive definite, thus nonsingular. H3 H4 So that in the process of decomposition, instead of going through Theorem 7, we just apply the LDU factorization to H, and get

−1 I 0 H1 0 I H1 H2 H = − − , H H 1 I 0 H T 0 I 3 1 1 −1 −1 where H3H1 and H1 H2 are symmetric matrices. Moreover, because of the symmetry of H, −1 −1 T −1 H3H1 = (H1 H2) = H1 H2. Therefore, every positive deﬁnite symplectic matrix can be written as I 0 P 0 I S H = (7) S I 0 P −T 0 I with P symmetric positive deﬁnite. Since P is already symmetric here, we do not have to decompose it into the product of two symmetric matrices, and just have the factorization P 0 I 0 I P − I I 0 I P −1 − I = , 0 P −T −P −1 I 0 I I I 0 I as a consequence I 0 I 0 I P − I I 0 I P −1 − I I S H = S I −P −1 I 0 I I I 0 I 0 I I 0 I P − I I 0 I S + P −1 − I = , S − P −1 I 0 I I I 0 I thus H ∈L4. I I I 0 I I On the other hand, J = ∈ L ⊂ L while J∈ / SPP , hence SPP ( 0 I −I I 0 I 3 4 L4.

15 Remark 6. The above factorization starts with an upper triangular matrix in right-hand side. Similar to the proof of Theorem 9 , we can use the LDU factorization in the case of H4 invertible, I S P 0 I 0 H = , (8) 0 I 0 P −T S I to obtain another decomposition where the ﬁrst one is a lower triangular matrix.

Going a step further, we can prove that L4 is optimal, i.e., SPP * L3.

Theorem 10. SPP * L3. 2I 0 Proof. Let G = ∈ SPP . If there exist S ,S ,S such that 0 1 I 1 2 3 2 I S I 0 I S I + S S S + S S S + S G = 3 1 = 3 2 1 3 2 1 3 , 0 I S I 0 I S I + S S 2 2 2 1 then we have S2 = 0, I =2I, which is a contradiction.

3.3 Symplectic M-matrices M-matrices occur very often in a wide variety of areas including finite difference methods for partial differential equations, economics, probability and statistics [6]. There are many equivalent definitions of M-matrices, we adopt the following one. n×n Definition 4. A ∈ R is an M-matrix if aij ≤ 0 for i =6 j and Re(λ) > 0 for every eigenvalue λ of A. The next result is a factorization of the symplectic M-matrices. It appears in [12] and is the key result on how symplectic M-matrices are decomposed into the product of several unit triangular symplectic matrices. Theorem 11. The set of 2d-by-2d symplectic M-matrices is × D ∈ Rd d positive diagonal T  I 0 D 0 I K H = H ≤ 0  SP M = ,  −1 T   H I 0 D 0 I K = K ≤ 0   

HDK diagonal     where the inequalities H ≤ 0 and K ≤ 0 mean that hij ≤ 0 and kij ≤ 0 for all i, j. Here we treat D as an ordinary symmetric matrix, as in Theorem 9, any symplectic M-matrix can be decomposed as the product of four triangular blocks.

Theorem 12. SP M ( L4. Thus any symplectic M-matrix can be factored into no more than 4 unit triangular symplectic matrices.

Proof. It is easy to show SP M ⊂ L4 based on Theorem 11. Moreover, −I2d ∈ L4 while −I2d ∈/ SP M, hence SP M ( L4.

Theorem 13. SP M * L3. Proof. It is similar to the proof of Theorem 10.

16 4 Unconstrained parametrization and optimization

In this section, we first consider the unconstrained parametrization of the symplectic matrices or its subsets. For a set Ω, we expect to find a map φ : Rn → Ω which is smooth and surjective. Therefore one may be able to deal with some problems on unconstrained parameter space Rn instead of Ω by employing φ. In the last subsection, we turn to the topic about how the unconstrained parametrization impacts on the optimization problems with symplectic constraints. In order to represent the free parameters of a symmetric matrix, we define the map P a for extracting the lower d×d T triangular parameters as P a(S)=(s11,s21,s22,s31, ··· ,sdd), where S =(sij) ∈ R , S = S.

4.1 Parametrization of general symplecic matrices

d×d Based on Theorem 6, one can parametrize SP eﬃciently. Take symmetric S1,S2, ··· ,S9 ∈ R , then I S I 0 I S H(P a(S ), ··· , P a(S )) = 9 ··· 1 (9) 1 9 0 I S I 0 I 2 can represent any symplectic matrix when P a(S1), ··· , P a(S9) vary. Note that here P a(Si) are 9d(d+1) free parameters without any constraint, thus H is a surjective map from R 2 into SP . With (9), it is easy to write out the inverse element and the transpose of H, i.e.,

− I −S I 0 I −S H(P a(S ), ··· , P a(S )) 1 = 1 ··· 9 , (10) 1 9 0 I −S I 0 I 8 I 0 I S I 0 H(P a(S ), ··· , P a(S ))T = ··· 8 . (11) 1 9 S I 0 I S I 1 9 The concise expressions of the inverse element and the transpose make it possible to construct more complex structured symplectic matrices.

4.2 Parametrization of positive deﬁnite symplectic matrices

By Theorem 9, we know that SPP is a subset of L4. Since Ln cannot guarantee the symmetry for its elements, a new collection of symplectic matrices is needed. The following theorem will show how to construct the unconstrained parametrization of SPP . Denote

2 T Ln = {LL |L ∈Ln}. By (10) and (11), we know L is invertible and LT is still symplectic, so that LLT is a positive 2 2 definite symplectic matrix. Thus Lm ⊂Ln ⊂ SPP for all integers 1 ≤ m ≤ n. 2 Theorem 14. SPP = L4. Thus any positive definite symplectic matrix can be represented as the product of a matrix which can be factored into no more than 4 unit triangular symplectic matrices and its transpose. Proof. According to (7) in the proof of Theorem 9, every positive definite symplectic matrix can be written as I 0 P 0 I S H = S I 0 P −T 0 I 17 with P symmetric positive definite. Let P 1/2 be the unique positive definite square root of P . P = P 1/2P 1/2 implies that P −1 =(P 1/2)−1(P 1/2)−1, then

I 0 P 1/2 0 P 1/2 0 I S H = S I 0 (P 1/2)−1 0 (P 1/2)−1 0 I T I 0 P 1/2 0 I 0 P 1/2 0 = . S I 0 (P 1/2)−1 S I 0 (P 1/2)−1 Just like what we did in Theorem 9, taking into account Lemma 2 and the symmetry of P 1/2, we have I 0 P 1/2 0 S I 0 (P 1/2)−1 I 0 I 0 I P 1/2 − I I 0 I (P 1/2)−1 − I = S I −(P 1/2)−1 I 0 I I I 0 I I 0 I P 1/2 − I I 0 I (P 1/2)−1 − I = , S − (P 1/2)−1 I 0 I I I 0 I hence we have obtained the expression of L exactly.

Based on Theorem 14, one can represent SPP by parameterizing L4. Take symmetric S1,S2,S3,S4 ∈ Rd×d, then denote

T H(P a(S1), ··· , P a(S4)) = LL , I 0 I S I 0 I S L = 3 1 . S I 0 I S I 0 I 4 2 2d(d+1) Here P a(Si) are free parameters without any constraint, thus H is a surjective map from R into SPP .

4.3 Parametrization of singular symplectic matrices There have been some studies on the structure of singular symplectic matrices [25, 26]. Deﬁnition 5. A symplectic matrix H ∈ SP is singular if det(H − I)=0. Denote all the singular symplectic matrices by

SPS = {H ∈ SP |H singular}.

A symplectic matrix H is singular when 1 is one of its eigenvalues, and there is a 0 =6 u ∈ R2d such that Hu = u where u is called the ﬁxed vector of H. Theorem 6 prompts us to consider the 2d following question: If there are several linearly independent vectors u1,u2, ..., un ∈ R , n ≤ 2d, how to search for a symplectic matrix H such that Hui = ui? Let us consider a slightly simpler case. Denote

V × × U = [u ,u , ··· ,u ]= 1 ∈ R2d n, V ,V ∈ Rd n, 1 2 n V 1 2 2 18 where U is a full-rank upper triangular matrix, and assume that rank(V1) = r1, rank(V2) = r2. Now let

I 0 Or1 0 (d−r1)×(d−r1) V1 = S = , T ∈ R symmetric , ( S I 0 T )

I S Or2 0 (d−r2)×(d−r2) V2 = S = , T ∈ R symmetric , ( 0 I 0 T )

VU = {Hk ··· H 2H1|Hi ∈V1 ∪V2,k ≥ 1}.

Note that r1 = n, r2 =0 when 1 ≤ n

Given any H ∈VU , it is easy to verify that HU = U. For instance, if H ∈V1, then

I 0 V1 V1 V1 HU = Or1 0 = Or1 0 = = U.  I V2  V1 + V2 V2 0 T 0 T     Note that the last d − r1 rows of V1 are all 0. Denote the set of all the symplectic matrices H satisfying HU = U by WU . It is easy to see that WU and VU are both groups, moreover, VU is a subgroup of WU . Whether VU = WU or VU ( WU is an interesting problem to be concerned with. At least, in the case of n = 0, Theorem 6 immediately points out that VU = WU since both of VU and WU equal to SP , while in the case of n =2d, VU and WU degenerate both into {I2d}. A theorem is proposed as follows.

2d×n Theorem 15. VU = WU . Here U ∈ R (1 ≤ n ≤ 2d) is any full-rank upper triangular matrix. Furthermore,

V = {H ··· H H |H − ∈V ,H ∈V } when 1 ≤ n

U11 U12 U11 U12 U = = U22 0 U22  0  0  e  d×d d×(n−d) d×(n−d) (n−d)×(n−d) where U11 ∈ R , U12 ∈ R , U22 ∈ R , U22 ∈ R . Since U is full-rank upper A1 B1 triangular, U11 and U22 are nonsingular upper triangular matrices. For any H = ∈ WU , e A2 B2 e A B U U U U HU = U ⇒ 1 1 11 12 = 11 12 A B 0 U 0 U 2 2 22 22 A U A U + B U U U ⇒ 1 11 1 12 1 22 = 11 12 ⇒ A = I, A =0. A U A U + B U 0 U 1 2 2 11 2 12 2 22 22 19 I B With A = I, A = 0 and the third item of Property 1, we ﬁnd that H = 1 . Moreover, 1 2 0 I

B11 B12 U22 A1U12 + B1U22 = U12 ⇒ B1U22 =0 ⇒ =0 B21 B22 0 e B11U22 ⇒ =0 ⇒ B11 =0, B21 =0, "B21U22# e and the symmetry of B1 showse that B12 = B21 = 0. Hence

O − 0 I n d H = 0 B ∈V ⊂V , 22 2 U 0 I    which means WU ⊂VU , consequently VU = WU . 2. Case 1 ≤ n

U1 U × × U = 1 = 0 , U ∈ Rd n, U ∈ Rn n, 0   1 1 e0   e A ⋆ where U is nonsingular upper triangular. For any H = ∈ W , 1 B ⋆ U e A A 11 12 ⋆ U U A ⋆ U U A A 1 1 HU = U ⇒ 1 = 1 ⇒  21 22  0 = 0 B ⋆ 0 0 B11 B12     ⋆ e0 e0  B B   21 22        A U A11 A12 U1 11 1 · U1 U1 A21 A22 0 "A21U1# ⇒   = 0 ⇒ e = 0 B11 B12 Ue     · 1 e0  B11U1  e0  B21 B22 0   e      "B21U1#    e   e  ⇒ A11 = In, A21 =0, B11 = On, B21 =0 .  e Property 1 leads to

A A T B B B B T A A 11 12 11 12 = 11 12 11 12 A A B B B B A A 21 22 21 22 21 22 21 22 I A T O B O B T I A ⇒ n 12 n 12 = n 12 n 12 0 A 0 B 0 B 0 A 22 22 22 22 O B O 0 ⇒ n 12 = n 0 AT B + AT B BT BT A + BT A 12 12 22 22 12 12 12 22 22 T T ⇒ B12 =0, A22B22 = B22A22.

20 Now we have A A I A 11 12 ⋆ n 12 ⋆ A21 A22 0 A22 T T H = = , A B22 = B A22.  B B   O 0  22 22 11 12 ⋆ n ⋆  B B   0 B   21 22   22      A A A Since rank( ) = d, we know rank( 22 ) = d − rank( I A ) = d − n, thus G = 22 B B n 12 B 22 22 is full-rank. Property 6 tells us that G is a part of a 2(d− n)-by-2( d − n) symplectic matrix A ⋆ G = 22 . Similar to the proof of Theorem 7, there exist a symmetric S and nonsingular P, Q B22 ⋆ such that e P 0 Id−n 0 Q 0 G11 ⋆ −T G −T = 0 P S I − 0 Q I − ⋆ d n d n where G11 is symmetric. We have e

In 0 I 0 On −A12Q 0 I T T 0 P On 0 −Q A12 Id−n − G11    I   In 0 0 −I − 0 I 0 − d n  0 P T        I A I 0   I 0 n 12 ⋆ n 0 0 A22 0 Q · On 0      I On 0 In 0 0 S ⋆ 0 −T  0 B22   0 Q           In A12Q I 0 On −A12Q ⋆ I T T 0 G11 = On 0 −Q A12 Id−n − G11    I   On 0 0 −Id−n 0 I ⋆  0 Id−n        I 0   I 0 n ⋆ 0 Id−n I ⋆ I T = On 0   = =  I On 0 0 ⋆ 0 I 0 −Id−n ⋆  0 Id−n        where T is symmetric, hence

In 0 I 0 −1 0 On A12Q 0 P I T T H = On 0   Q A12 G11 − Id−n  I In 0   0 −S 0 0 I  0 P T         I 0  I 0 n 0 I T 0 Q−1 · On 0   .  I 0 I In 0 0 Id−n 0  0 QT        Taking into account Remark 2 and Lemma 2, we can easily factor H into 10 unit triangular symplectic matrices from V1 and V2. Therefore WU ⊂VU , consequently WU = VU .

21 Now return to the topic about how to parameterize singular symplectic matrices. Theorem 15 gives a speciﬁc structure of singular symplectic matrices with many linearly independent vectors as T their eigenvectors, especially when U = e1 = [1, 0, ··· , 0] , we obtain We1 = Ve1 whose structure is clear to be freely parameterized.

Corollary 6. The set of 2d-by-2d singular symplectic matrices is

I 0 I 0 I S I S − SPS = Q 0 0 9 ··· 0 0 1 Q 1  I 0 I  I 0 I ( 0 S10 0 S2     d×d (d−1)×(d−1) S2i−1 ∈ R symmetric, S2i ∈ R symmetric, Q symplectic . )

One can express Q as the product of 9 unit triangular symplectic matrices if needed. I 0 I S Proof. For any H = Q 0 0 ··· 1 Q−1 ∈ SPS, let u = Qe =6 0. Then it is easy to  I 0 I 1 0 S10 check that Hu = u, hence H is singular. For any singular H ∈ SP , assume that Hu = u where 0 =6 u ∈ R2d. From Property 5 and −1 Property 6, we know that u is a part of a symplectic matrix Q = [u ⋆]. Therefore Q HQe1 = −1 −1 −1 Q Hu = Q u = e1, which means that Q HQ ∈ We1 = Ve1 due to Theorem 15. We immediately know that H ∈ SPS.

4.4 Unconstrained optimization An optimization problem with symplectic constraint is in the following form

min f(X), s.t. XT JX = J. (12) X∈R2d×2d

Because of the antisymmetry of XT JX, the constraint of the problem (12) is made up of 2d2 − d equations. There have been many works on optimization on the real symplectic group [7, 15, 34], in which one performs optimization by considering the gradients along the manifold. In this work, the unit triangular factorization provides an approach to the symplectic optimization from a new perspective, i.e., optimizing in a higher dimensional unconstrained parameter space. By (9), problem (12) is equivalent to

min f(H(P a(S1), ··· , P a(S9))), d(d+1) P a(Si)∈R 2 which is indeed an unconstrained optimization problem. In recent years, with the expansion of variables required by practical application, the limitations of existing constrained optimization methods are reﬂected in practice. Fortunately, the unconstrained parametrization of the symplectic matrices allows us to circumvent this limitation and apply faster and more eﬃcient unconstrained optimization algorithms, such as the recently popular deep learning techniques, which are able to deal with the optimization problems with billions of unconstrained parameters. In fact, this method

22 has been utilized in our recent work [23], where we construct symplectic neural networks based on the factorization, and achieve a great success. In such a case, the unit triangular factorization- based optimization can be implemented directly within the deep learning framework and performs well, while the traditional Riemannian-steepest-descent approach faces challenges.

5 Conclusions

The factorization theorems shown in this work overcomes some defects of conventional factorizations. [9, 12, 21, 31, 36] provide several factorizations of the matrix symplectic group. However, all the factorizations require cells of symplectic interchanges, permutation matrices or symplectic- orthogonal matrices, which are not elementary enough hence hard to be freely parameterized. [16, 27] factor the 2d-by-2d symplectic matrices as the products of at most 4d symplectic transvections, which can freely parameterize the matrix symplectic group, nevertheless, may be unstable with a large d in practice. Theorem 6 proves that any symplectic matrix can be factored into no more than 9 unit triangular symplectic matrices. The core of the proof is Theorem 7, which writes a symplectic matrix as the product of a symplectic matrix with nonsingular left upper block and a unit triangular symplectic matrix. Subsequently, we derive some important corollaries by Theorem 6 and Theorem 7, such as, (i) the determinant of symplectic matrix is one, (ii) the matrix symplectic group is path connected, (iii) all the unit triangular symplectic matrices form a set of generators of the matrix symplectic group. Furthermore, this factorization yields eﬀective methods for unconstrained parametrization of the matrix symplectic group and its structured subsets. With the unconstrained parametrization, we are able to apply the unconstrained optimization algorithms to the problems with symplectic constraints. Although we proved that any symplectic matrix can be factored into 9 unit triangular symplectic matrices, it is still unknown that whether “9” is the optimal number in this factorization. What we know regarding to the optimal number is that it is indeed between 4 and 9. This problem is left for future work.

Acknowledgments

This research is supported by the Major Project on New Generation of Artiﬁcial Intelligence from MOST of China (Grant No. 2018AAA0101002), and National Natural Science Foundation of China (Grant No. 11771438).

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