辛矩陣與矩陣對之分類

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(1)國立臺灣師範大學數學系碩士班碩士論文. 指導教授：謝世峰博士. 辛矩陣與矩陣對之分類 The Classification of Symplectic Matrices and Pairs. 研究生：游逸翔. 中華民國 106 年 7 月.

(2) 誌. 謝. 能完成這份論文，首先要感謝我的指導教授謝世峰老師的協助與教導，謝老師不僅在課業上也在生活上給予我極大的幫助。謝謝口試委員郭岳承老師為本篇論文提供許多重要想法，並且仔細審閱，讓論文內容更臻完善。謝謝口試委員黃聰明老師為本篇論文提供的意見與指導，讓我思考未曾深入探討的部分。我必須感謝我的好友許勝溢，因為他的鼓勵與指導，我才得以進入研究所提升自己。也要謝謝張閔翔，和我一起討論論文內容。謝謝所有在研究所期間給我各種指導的各位老師，讓我無論在數學的專業知識或教學能力上，比起從前都有大大的提升。還要謝謝研究所的各位同學，不僅在課業上能夠彼此討論支持，也讓研究所生活增添許多趣味。最後更要感謝我的家人，我的太太善霖常常必須很辛苦的一個人帶著兩個小孩，謝謝善霖在我讀研究所的這段期間極其辛勞的付出與支持。謝謝我的父親游金河先生與母親紀秀津女士，在我人生中各個階段的參與，謹將這篇論文，獻給他們。.

(3) The Classification of Symplectic Matrices and Pairs Yi-Siang You. Contents Abstract. 1. 1 Introduction. 2. 2 Preliminaries. 4. 3 The Classification of Symplectic Matrices. 13. 4 The Classification of Symplectic Pairs. 19. 5 Conclusion and Future Works. 29. 6 Refrences. 29 Abstract. In applications a symplectic matrix is often required to be partitioned with a nonsingular block. By applying the complementary bases theorem of Dopico and Johnson in [3], we can rearrange a symplectic matrix with a swap matrix to obtain a nonsingular block. We classify symplectic matrices with corresponding swap matrices. Moreover, a rearrangement of symplectic pair by Mehrmann and Poloni in [8] merges a regular symplectic pair into a symplectic matrix. Therefore we can classify regular symplectic pairs with similar approach.. Key Words: symplectic matrix, symplectic pair, complementary bases theorem, hermitian matrix, Lagrangian subspace, minimal classification. 1.

(4) 1. Introduction. Symplectic matrices play an important role in classical mechanics and Hamiltonian dynamical systems [1, 2, 3]. Moreover, the symplectic matrices appear in the linear control theory for discrete-time systems [3, 4, 7]. A standard symplectic matrix Jn ∈ C2n×2n is a matrix in the partitioned form [ ] O I Jn = , −I O where I is the n × n identity matrix. In the following text, 2n × 2n matrices are always partitioened[in four n ×]n submatrices if not otherwise stated. That is, in the S11 S12 representation S = , S11 , S12 , S21 , and S22 are all in Cn×n . And the 2n×2n S21 S21 standard symplectic matrix will be simply denoted by J if the order is clear. With the standard symplectic matrix Jn we define symplectic matrices and symplectic pairs as following. Definition 1.1. A matrix S ∈ C2n×2n is symplectic if SJS ∗ = J, where S ∗ is the conjugate transpose of S. A matrix pair (A, B) with A, B ∈ C2n×2n is called a symplectic pair if AJA∗ = BJB ∗ . We denote the set of all 2n × 2n symplectic matrices by Sn . Definition 1.2. A symplectic λB) ̸= 0 for some λ ∈ C. We by SPn . Two symplectic pairs A1 = M A2 and B1 = M B2 for by. pair (A, B) with A, B ∈ C2n×2n is regular if det(A − denote the set of all 2n × 2n regular symplectic pairs (A1 , B1 ) and (A2 , B2 ) are said to be left equivalent if some nonsingular square matrix M ∈ C2n×2n , denoted (A1 , B1 ) ∼ (A2 , B2 ).. Most of the time we need the “regularity assumption” when a symplectic pair is mentioned. In this thesis we are going to classify symplectic matrices and regular symplectic pairs with “swap matrices”. The definition of swap matrix is as following. Definition 1.3. Let v, vb ∈ {0, 1}n such that vbj = 1 − vj . Then Πv ∈ C2n×2n with [ ] diag(b v ) diag(v) Πv = −diag(v) diag(b v) is called a swap matrix. We use Pn to denote the set of all 2n × 2n swap matrices. Remark 1.1. For example, let e = {1, 1, · · · , 1} and 0 = {0, 0, · · · , 0} be the ndimensional vectors whose entries are all 1’s and 0’s respectively. Then J = Πe and I = Π0 . In Proposition 2.8, we will show that every swap matrix is symplectic. 2.

(5) Definition 1.4. Let Π, Π1 , and Π2 ∈ Pn . We define

(6) { [ ] }

(7) S11 S12 .

(8) SΠ .= S ∈ Sn

(9) ΠS = , det S22 ̸= 0 S21 S22 and SPΠ1 ,Π2. ..=. { (A, B) ∈ SPn.

(10) ([ ] [ ] )

(11) X O I X 12 11 ∗

(12) (A, B) ∼ Π1 , Π2 , ∗

(13) X22 I O X12 [ ] } X11 X12 2n×2n for some Hermitian ∈C . ∗ X12 X22. The first main result is the classification of symplectic matrices indexed by Pn . The second main result of this thesis is the classification of regular symplectic pairs indexed by Pn × Pn . We state them as the two following theorems. Theorem 1.1 (Classification of Symplectic Matrices). (i) For each S ∈ Sn , there exists a swap matrix Π ∈ Pn and a Hermitian matrix ] [ X11 X12 ∈ C2n×2n X= ∗ X22 X12 such that. [ ΠS =. (ii) Sn =. ∪ Π∈Pn. I X11 ∗ O X12. ]−1 [. X12 O X22 I. ] .. SΠ .. (iii) For each Π ∈ Pn , there exists S ∈ SΠ such that S ∈ / SΠ′ for all Π′ ̸= Π. (iv) For each Π ∈ Pn , SΠ is an open set relative to Sn . Theorem 1.2 (Classification of Regular Symplectic Pairs). (i) For each regular symplectic pair (A, B) ∈ SPn , there exist swap matrices Π1 , Π2 ∈ Pn and a Hermitian matrix [ ] X11 X12 X= ∈ C2n×2n ∗ X12 X22 ([. such that (A, B) ∼. X12 O X22 I. That is, (A, B) ∈ SPΠ1 ,Π2 . 3. [. ] Π∗1 ,. I X11 ∗ O X12. ]. ) Π2 ..

(14) (ii) SPn =. ∪ Π1 ,Π2 ∈Pn. SPΠ1 ,Π2 .. (iii) Each SΠ1 ,Π2 is an open set relative to SPn . To classify regular symplectic pairs, we have to introduce Lagrangian subspace which has close relationship with symplectic matrices. Definition 1.5. A Lagrangian subspace U is an n-dimensional subspace of C2n such that u∗ Jv = 0 for each u, v ∈ U . By the definition of Lagrangian subspace, we can easily verify that if S = [S1 S2 ] ∈ C is symplectic, where S1 , S2 ∈ C2n×n , then the columns of S1 span a Lagrangian subspace. Conversely, if the columns of S ∈ C2n×n spans a Lagrangian subspace, we can construct a 2n × 2n symplectic matrix related to S. These properties, which mean a close relationship between Lagrangian subspaces and symplectic matrices, will be proved in Propositions 2.13 and 2.14. This thesis is organized as follows. In section 2 we introduce symplectic matrices, swap matrices, Lagrangian subspaces, and the complementary bases theorem, without this theorem we cannot classify the symplectic matrices this way. In section 3, based on the complementary bases theorem, we classify symplectic matrices with swap matrices Π ∈ Pn and show that this classification is a minimal classification. In section 4, by transforming a regular symplectic pair into a symplectic matrix, we apply similar method in section 3 to classify regular symplectic pairs. But we provide an example showing this classification is not minimal. Finally, some conclusions and problems are presented in section 5. 2n×2n. 2. Preliminaries. In this section, we shall give some preliminary results. First we introduce some properties of the standard symplectic matrix J. Proposition 2.1. J −1 = J ∗ = −J. [ ] [ ] O I O −I 2n×2n ∗ Proof. Let J = ∈ C . Then J = −J = ∈ C2n×2n . By −I O I O simple computation we have [ ][ ] [ ] O I O −I I O ∗ JJ = = . −I O I O O I. 4.

(15) Proposition 2.2. det J = 1. Proof. [ det J = det. O I −I O. ]. = −(det I)(det(−I)) = 1.. Proposition is nonsingular. Moreover, if we partition [ 2.3. Every ] symplectic[matrix ] ∗ ∗ S11 S12 S22 −S12 S as S = , then S −1 = . ∗ ∗ S21 S22 −S21 S11 Proof. Let S be symplectic then SJS ∗ = J. Hence det(SJS ∗ ) = det(S) det(J) det(S ∗ ) = det(J). Since det(S) and det(S ∗ ) are conjugate, we have | det(S) |= det(S) det(S ∗ ) = 1. Thus, S is nonsingular. We obtain the inverse of S by the following computation. S −1 = S −1 JJ −1 = S −1 (SJS ∗ )J −1 = JS ∗ (−J) ][ ] [ ][ ∗ ∗ O −I O I S11 S21 = ∗ ∗ S12 S22 I O −I O [ ] [ ] ∗ ∗ S12 S22 O −I = ∗ ∗ −S11 −S21 I O ] [ ∗ ∗ S22 −S12 . = ∗ ∗ −S21 S11. In fact, det S = +1 for each symplectic matrix S, whether the entries of S are complex or real. See [2, 3, 5, 6]. Proposition 2.4. If S is symplectic, then S ∗ , S −1 , and (S ∗ )−1 = (S −1 )∗ are also symplectic.. 5.

(16) Proof. To show that S ∗ and S −1 are symplectic, we simply need to verify if these two matrices satisfy the definition given in Definition 1.1. First note that J = SJS ∗ ⇐⇒ ⇐⇒ ⇐⇒ ⇐⇒ ⇐⇒. S −1 JS ∗−1 = J S −1 (−J)S ∗−1 = −J S −1 J −1 S ∗−1 = J −1 (S ∗ JS)−1 = J −1 S ∗ JS = J.. The equivalence above shows that S ∗ is symplectic. Similarly, we also have S ∗ JS = J ⇐⇒ ⇐⇒ ⇐⇒ ⇐⇒. S ∗ (−J)S = −J S ∗ J −1 S = J −1 (S −1 JS ∗−1 )−1 = J −1 S −1 J(S −1 )∗ = J.. It follows that S is symplectic if and only if S ∗ or S −1 is symplectic. Therefore, by combining consequences above we have (S ∗ )−1 = (S −1 )∗ is also symplectic. Proposition 2.5. If S1 and S2 are symplectic, then S1 S2 is symplectic. Proof. (S1 S2 )J(S1 S2 )∗ = S1 (S2 JS2∗ )S1∗ = S1 JS1∗ = J. Remark 2.1. From Propositions 2.4 and 2.5, we see that Sn forms a group. After introducing propositions of symplectic matrices, we shall introduce some propositions of swap matrices, with which we classify symplectic matrices. In the following text, we denote the j-th column of the identity matrix by ej , the j-th column of a matrix S by Cj (S), the j-th row of a matrix S by Rj (S), and the positive number set {1, 2, 3, · · · , n} by [n]. Lemma 2.6. Let v ∈ {0, 1}n . Then [diag(v)] [diag(b v )] = [diag(b v )] [diag(v)] = O, [diag(v)] [diag(v)] = [diag(v)] , and [diag(b v )] [diag(b v )] = [diag(b v )] . Proof. Let α ⊂ [n] such that vj = 1 for j ∈ α and vj = 0 for j ∈ / α. Then { ∗ ej , for j ∈ α, Rj ([diag(v)]) = 0, for j ∈ / α, { 0, for j ∈ α, Rj ([diag(b v )]) = / α, e∗j , for j ∈ 6.

(17) Hence in [diag(v)] [diag(b v )], if j ∈ / α, Rj ([diag(v)]) = 0 will turn the corresponding Rj ([diag(b v )]) = e∗j into 0 row. If j ∈ α, Rj ([diag(v)]) = e∗j will keep the corresponding Rj ([diag(b v )]) = 0 as 0 row. So we have [diag(v)] [diag(b v )] = O. Similarly, [diag(b v )] [diag(v)] = O. As to the other two equations, we have [diag(v)] [diag(v)] = [diag(v)] (I − [diag(b v )]) = [diag(v)] and [diag(b v )] [diag(b v )] = [diag(b v )] (I − [diag(v)]) = [diag(b v )] . ∗ Proposition 2.7. If Πv ∈ Pn , then Π−1 v = Πv .. Proof. Let Πv ∈ Pn . By definition 1.3 of Πv , we see that [ ] [ ] diag(b v ) diag(v) diag(b v ) −diag(v) ∗ Πv = , Πv = , −diag(v) diag(b v) diag(v) diag(b v) and [diag(b v )] + [diag(v)] = In . Hence [ ][ ] diag(b v ) diag(v) diag(b v ) −diag(v) ∗ Πv Πv = −diag(v) diag(b v) diag(v) diag(b v) [ ] diag(b v )diag(b v ) + diag(v)diag(v) −diag(b v )diag(v) + diag(v)diag(b v) = −diag(v)diag(b v ) + diag(b v )diag(v) diag(v)diag(v) + diag(b v )diag(b v) = I.. With the above proposition, we can prove that every swap matrix is symplectic. Proposition 2.8. Every swap matrix is a symplectic matrix. Proof. Since [ ∗ Πv JΠv = [ = [ = [ =. ][. ][ ] O I diag(b v ) −diag(v) −I O diag(v) diag(b v) ][ ] −diag(v) diag(b v) diag(b v ) −diag(v) −diag(b v ) −diag(v) diag(v) diag(b v) diag(b v ) diag(v) −diag(v) diag(b v). −diag(v)diag(b v ) + diag(b v )diag(v) diag(v)diag(v) + diag(b v )diag(b v) −diag(b v )diag(b v ) − diag(v)diag(v) diag(b v )diag(v) − diag(v)diag(b v) ] O I , −I O. Πv satisfies Definition 1.1 of symplectic matrix. 7. ].

(18) Besides the swap matrices, we introduce two more types of symplectic matrices. These two types of symplectic matrices will help us to show the complementary bases theorem. n×n Proposition 2.9. If P ∈ Cn×n is matrix A∈C is a nonsingular [ a permutation ] [ and ] ∗−1 P O A O matrix, then the matrices SP ..= and SA ..= are also symplectic. O P O A. Proof. The assertion simply follows the definition of symplectic matrix. Since [ ][ ][ ∗ ] P O O I P O ∗ SP JSP = O P −I O O P∗ [ ][ ∗ ] O P P O = −P O O P∗ [ ] O PP∗ = −P P ∗ O [ ] O I = −I O =J and. [. AP JA∗P. ] [ −1 ] O I A O = −I O O A∗ [ ] [ −1 ] O A∗−1 A O = −A O O A∗ [ ] O A∗−1 A∗ = −AA−1 O [ ] O I = −I O A∗−1 O O A. ][. = J, the assertion of this proposition follows. The following theorem, the complementary bases theorem, is mainly quoted from [3]. We made some modifications to fit our goal here. To make the statements more succinct, we define some notations before introducing this theorem. Let α, β ⊂ [n]. Rα (S) and Cβ (S) denote the rows of S indexed by α and the columns of S indexed by β, respectively. If the row vectors of Rα (S) or the column vectors of Rα (S) are linearly independent and |α| = rank(S), then we call Rα (S) or Cα (S) maximal linearly independent respect to S. We use MI(S) to denote a collection of such sets. 8.

(19) [. ] S11 S12 Theorem 2.10 (Complementary Bases Theorem). Let S = ∈ Sn , α ⊂ S21 S22 [n], and α′ = [n]\α. If Rα (S22 ) ∈ MI(S22 ), then the the rows of Rα (S22 ) and Rα′ (S12 ) together constitute an n × n nonsingular matrix. [ ] S11 S12 Proof. Given that S = ∈ Sn and Rα (S22 ) ∈ MI(S22 ), suppose that S21 S22 ′ rank(S22 ) = k, α = {α1 , α2 , · · · , αk }, and α′ = {α1′ , α2′ , · · · , αn−k }, where αi < αi+1 ′ ′ and αi < αi+1 . The first step is to put the linearly independent rows of Rα (S22 ) together by permutation. Let P = [pij ] ∈ Cn×n be a permutation matrix defined by { ′ 1, for j = αi or j = αi−k , pij = 0, otherwise. This permutation will move the rows of Rα (S22 ) to the first k positions and[ the rows ] P O of Rα′ (S12 ) to the last (n − k) positions. By Propositions 2.5 and 2.9, SP = O P ] [ P S11 P S12 . We see that R[k] (P S22 ) ∈ is symplectic and so is SP S. Let SP S = P S21 P S22 MI(P S22 ). Now our goal is to show that the rows of R[k] (P S22 ) and R[k]′ (P S12 ) together constitute a nonsingular matrix. The second step is to make the submatrices into a row-reduced echelon form. That form is simpler for our observation. Before doing so, let ] ] [ ′ [ ∗ ∗ ′ ∗ S11 S12 P∗ S11 P S21 ∗ ′ .. . = S = (SP S) = ′ ′ ∗ ∗ S22 S21 P∗ P ∗ S22 S12 ′ ′ Then C[k]′ (S21 ) = (R[k]′ (P S12 ))∗ and C[k] (S22 ) = (R[k] (P S22 ))∗ . Since the conjugate ′ ′ ). Note that transpose does not change the linear independency, C[k] (S22 ) ∈ MI(S22 ′ ′ the goal now is to show that the columns of C[k]′ (S21 ) and C[k] (S22 ) constitute an n × n nonsingular matrix. ′ Let Y ∈ Cn×n be a nonsingular matrix [such that Y]S22 is a row-reduced echelon ∗−1 Y O matrix. By Propositions 2.5 and 2.9, SY = is symplectic, therefore so is O Y SY S ′ . Then we have ] [ ∗−1 ][ ′ ′ Y O S11 S12 ′ SY S = ′ ′ S21 S22 O Y [ ∗−1 ′ ] ′ Y S11 Y ∗−1 S12 = . (2.1) ′ ′ Y S21 Y S22. 9.

(20) Since multiplication does not change the linear independency, now the goal is to show ′ ′ that the columns of C[k]′ (Y S21 ) and C[k] (Y S22 ) form a[nonsingular ] matrix. Note that in Ik Z ′ ′ (2.1) Y S22 is of row-reduced echelon form. Y S22 = for some Z ∈ Ck×(n−k) O O ] [ X11 X12 k×k ′ conformally with and identity matrix Ik ∈ C . Partition Y S21 = X21 X22 X11 ∈ Ck×k and X22 ∈[C(n−k)×(n−k) to show]that X22 is nonsingular. ] . Now [ it∗−1suffices ′′ ′′ ′ ′ S11 S12 Y S11 Y ∗−1 S12 Let S ′′ = SY S ′ = = . Since S ′′ is symplectic, ′′ ′′ ′ ′ S21 S22 Y S Y S 22] [ 21′′∗ ′′∗ S −S 22 12 by Proposition 2.3 we know that S ′′−1 = ′′∗ ′′∗ . Then we have −S21 S11 [ ′′ ][ ] [ ] ′′ ′′∗ ′′∗ S11 S12 S22 −S12 I O ′′ ′′−1 S S = = . ′′ ′′ ′′∗ ′′∗ S21 S22 −S21 S11 O I Thus ′′ ′′∗ ′′ ′′∗ O = S21 S22 − S22 S21 ′ ′ ∗ ′ ′ ∗ = Y S21 (Y S22 ) − Y S22 (Y S21 ) ][ ] [ ][ ∗ ] [ ∗ Ik O Ik Z X11 X21 X11 X12 − = ∗ ∗ X21 X22 Z∗ O O O X12 X22 [ ] ∗ ∗ ∗ ∗ X11 + X12 Z ∗ − X11 − ZX12 −X21 − ZX22 = X21 + X22 Z ∗ O. and X21 = −X22 Z ∗ . Since S ′′ is nonsingular, any subset[of rows of S ′′ is linearly independent. In par] X21 (X22 [O O are ticular, all the rows of([the submatrix ]) ]) linearly independent. ∗ Hence n − k = rank X21 X22 = rank X22 −Z I ≤ rank (X22 ) . Since (n−k)×(n−k) X22 ∈ C , we conclude that X22 is nonsingular and thus the claim follows. The complementary bases theorem is more general than what we have proved. In fact, the statement is still true for other submatrices and for columns. We shall prove this more general version of the complementary bases theorem. Corollary [ ]2.11 (The General Version of Complementary Bases Theorem). Let S = S11 S12 ∈ Sn . If Rα (Sij ) ∈ MI(Sij ), then the rows of Rα (Sij ) and Rα′ (Si′ j ), S21 S22 together constitue an n × n nonsignular matrix. Similarly, if Cα (Sij ) ∈ MI(Sij ), then the columns of Cα (Sij ) and Cα′ (Sij ′ ) together constitue an n × n nonsignular matrix. (The notation we used above: if i = 1 then i′ = 2. If i = 2 then i′ = 1. j is similar.) 10.

(21) [. ] S11 S12 Proof. Let S = ∈ Sn . We first prove the three other row versions of S21 S22 complementary bases theorem. Since Jn ∈ Sn , by Proposition 2.5 we know all the three following matrices, [ ] −S12 S11 SJ = , −S22 S21 [ ] S21 S22 JS = , and −S11 −S12 [ ] −S22 S21 JSJ = S12 −S11 are symplectic. Since the sign makes no difference of the linear independence, by Theorem 2.10, for i, j ∈ {1, 2}, if Rα (Sij ) ∈ MI(Sij ), then the rows of Rα (Sij ) and Rα′ (Si′ j ) constitute a nonsingular n × n matrix. Now we prove [the column]version of the complementary bases theorem. By propo∗ ∗ S21 S11 ∗ is symplectic. Hence by Theorem 2.10, if Rα (S22 ) ∈ sition 2.4, S ∗ = ∗ ∗ S12 S22 ∗ ∗ ∗ ) constitute a nonsingular matrix. Since ) and Rα′ (S21 ), then the rows of Rα (S22 MI(S22 ∗ ∗ ∗ ∗ Rα (S22 ) = (Cα (S22 )) , Rα′ (S21 ) = (Cα′ (S21 )) , and the conjugate transpose does not change linear independency, we conclude that if Cα (S22 ) ∈ MI(S22 ), then the columns of Cα (S22 ) and Cα′ (S21 ) constitute a nonsingular matrix. By similar method, we can prove the other three column cases. After introducing symplectic matrix and its propositions, now we introduce some relative properties about Lagrangian subspace. [ ] U1 Proposition 2.12. Let U = ∈ C with U1 , U2 ∈ Cn×n . The columns of span a U2 Lagrangian subspace if and only if U1∗ U2 = U2∗ U1 . [ ] U1 Proof. The columns of U = span a Lagrangian subspace of C2n , by Definition U2 1.5 of Lagrangian subspace, if and only if u∗ Jv = 0 for each column vector u, v of U [ ]∗ [ ][ ] U1 O I U1 ⇐⇒ =O U2 −I O U2 ⇐⇒ − U2∗ U1 + U1∗ U2 = O ⇐⇒ U1∗ U2 = U2∗ U1 .. 11.

(22) To show the relationship between a basis of Lagrangian subspace and a symplectic matrix, we give the following definition. [ ] U1 Definition 2.1. Let U = with U1 , U2 ∈ Cn×n . If the columns of U span a U2 [ ] U1 −U2 2n . Lagrangian subspace of C , we define SU .= . U2 U1 [ ] Q1 Proposition 2.13. If Q = where Q1 , Q2 ∈ Cn×n is orthogonal and if the Q2 [ ] Q1 −Q2 2n columns of Q span a Lagrangian subspace of C , then SQ = is orthogonal Q2 Q1 and symplectic. ] [ Q1 Proof. If Q = ∈ C2n×n is orthogonal, then Q2 [ ] ] ] Q1 [ [ ∗ ∗ (2.2) = Q∗1 Q1 + Q∗2 Q2 . In = Q1 Q2 Q2 [ ] Q1 Recall that Proposition 2.12 shows that if the columns of Q = spans a LaQ2 grangian subspace, then Q∗1 Q2 = Q∗2 Q1 . By some computation, equations (2.2) and Proposition 2.12 imply that [ ][ ][ ] Q∗1 Q∗2 O I Q1 −Q2 −Q∗2 Q∗1 −I O Q2 Q1 ] ][ [ ∗ ∗ Q2 Q1 Q1 Q2 = ∗ ∗ −Q1 Q2 −Q2 Q1 ] [ ∗ ∗ Q∗1 Q1 + Q∗2 Q2 Q1 Q2 − Q2 Q1 = −Q∗2 Q2 − Q∗1 Q1 −Q∗2 Q1 + Q∗1 Q2 [ ] O I = . −I O ] [ ] [ Q1 −Q2 Q∗1 Q∗2 and By Definition 1.1 and Proposition 2.4 we have that both Q2 Q1 −Q∗2 Q∗1. 12.

(23) are symplectic. To show that SQ is orthogonal, we simply compute that [ ][ ] Q∗1 Q∗2 Q1 −Q2 ∗ SQ SQ = −Q∗2 Q∗1 Q2 Q1 [ ] ∗ ∗ Q1 Q1 + Q2 Q2 −Q∗1 Q2 + Q∗2 Q1 = −Q∗2 Q1 + Q∗1 Q2 Q∗2 Q2 + Q∗1 Q1 [ ] I O = . O I [ Proposition 2.14. If S =. S11 S12 S21 S22. ]. [ ∈ Sn , then the columns of. S11 S12. ] span a. Lagrangian subspace. Proof. Since S ∈ Sn , by Proposition 2.4, S ∗ ∈ Sn . Hence have ] [ ∗ ∗ ∗ ∗ −S21 S11 + S11 S21 −S21 S12 + S11 S22 ∗ = J. S JS = ∗ ∗ ∗ ∗ S22 S12 + S12 S21 −S22 S11 + S12 −S22 [ Hence. ∗ S11 −S21. +. ∗ S21 S11. = O. By Proposition 2.12, the columns of. S11 S12. ] span a. Lagrangian subspace.. 3. The Classification of Symplectic Matrices. In this section we classify the symplectic matrices with swap matrices. First we show that for each S ∈ Sn , there exists a swap matrix Π ∈ Pn such that ΠS = L−1 M where L ∈ C2n×2n is an upper triangle matrix and M ∈ C2n×2n is a lower ∪ triangle matrix. Based on this we classify the symplectic matrices. That is, Sn = Π∈Pn SΠ . According to the definition of swap matrix, we have 2n classes of symplectic matrices. We furthermore prove that this classification is minimal and that each SΠ is an open set in Sn in the end. [ ] S11 S12 Lemma 3.1. S = ∈ Sn with S22 being nonsingular, if and only if there S21 S22 [ ] X11 X12 exists a Hermitian matrix X = with X12 being nonsingular such that ∗ X12 X22 [ S=. I X11 ∗ O X12. ]−1 [. 13. X12 O X22 I. ] ..

(24) [. S11 Proof. Suppose that S = S21 exists. It is easy to verify that ][ [ −1 I −S12 S22 −1 O S22. S12 S22. ] −1 ∈ Sn with S22 being nonsingular, then S22. S11 S12 S21 S22. ]. [ =. −1 S11 − S12 S22 S21 O −1 S21 I S22. ] .. To prove this lemma, it suffices to show that the matrix ] [ −1 −1 −S12 S22 S11 − S12 S22 S21 −1 −1 S21 S22 S22 is Hermitian. Since J is symplectic, by Proposition 2.4, we see that SJS ∗ = S ∗ JS = J. For SJS ∗ = J, we have ] ] [ ][ ∗ [ ][ ][ ∗ ∗ ∗ −S12 S11 S11 S21 S11 S12 O I S11 S21 = ∗ ∗ ∗ ∗ S22 S22 −S22 S21 S12 S21 S22 −I O S12 ] [ ∗ ∗ ∗ ∗ + S11 S22 −S12 S21 + S11 S12 −S12 S11 = ∗ ∗ ∗ ∗ + S21 S22 −S22 S21 + S21 S12 −S22 S11 [ ] O I = . (3.1) −I O ∗ ∗ . Since S22 is nonsin= S22 S21 Noticing the 22-block of (3.1), it turns out that S21 S22 ∗ exsist. By multiplying the inverse matrices gular, the inverse of matrices of S22 and S22 on both sides we have ( −1 )∗ −1 ∗ ∗−1 S22 S21 = S21 S22 = S22 S21 . (3.2). In addition, for S ∗ JS = J, we have ] [ ][ ][ [ ∗ ∗ O I S11 S12 S11 S21 = ∗ ∗ S21 S22 S12 S22 −I O [ = [ =. ∗ ∗ −S21 S11 ∗ ∗ −S22 S12. ][. S11 S12 S21 S22. ]. ] ∗ ∗ ∗ ∗ S12 + S11 S22 S21 −S21 −S21 S11 + S11 ∗ ∗ ∗ ∗ S11 + S12 S21 −S22 S12 + S12 S22 −S22 ] O I . (3.3) −I O. −1 ∗ ∗ Notincing again the 22-block of (3.3) we see that S22 S12 = S12 S22 . By multiplying S22 ∗−1 and S22 on both sides we obtain ( ) −1 ∗−1 ∗ −1 ∗ S12 S22 = S22 S12 = S12 S22 . (3.4). 14.

(25) −1 ∗ ∗ Moreover, from the 12-block of (3.3) we have −S21 S12 + S11 S22 = I. Multiplying S22 −1 −1 ∗ ∗ on both sides we have −S21 S12 S22 + S11 = S22 . Combining this with (3.4) we obtain −1 ∗ −1 ∗ −1 ∗ ∗ −1 ∗ ) = S11 − (S12 S22 ) S21 = S11 − S12 S22 S21 . S12 S22 (S22 ) = (S11 − S21. (3.5). From (3.2), (3.4), and (3.5), we conclude that ] [ ] [ −1 −1 X11 X12 S21 S11 − S12 S22 −S12 S22 = −1 −1 ∗ S21 X12 S22 X22 S22 −1∗ is Hermitian and X12 = S22 is nonsingular. [ ]−1 [ ] I X11 X12 O −1 Conversely, suppose that S = L M = for some HermiX I O X 21 22 ] [ X11 X12 tian matrix X = with X12 being nonsingular. By some computation we X21 X22 have [ ][ ][ ] I X11 O I I O ∗ LJL = ∗ ∗ O X21 −I O X11 X21 [ ][ ] −X11 I I O = ∗ ∗ −X21 O X11 X21 ] [ ∗ ∗ X21 −X11 + X11 = −X21 O [ ] O X12 = −X21 O. and. [. ][. ][ ∗ ] ∗ O I X12 X22 M JM = −I O O I ][ ∗ ] [ ∗ O X12 X12 X22 = −I X22 O I [ ] O X12 = ∗ ∗ −X12 −X22 + X22 [ ] O X12 = . −X21 O ∗. X12 O X22 I. Since LJL∗ = M JM ∗ and L−1 exists imply that L−1 M is symplectic, S = L−1 M is symplectic. 15.

(26) Lemma 3.2. [ ] If S ∈ Sn , then there exists a swap matrix Π ∈ Pn such that ΠS = ′ ′ S11 S12 ′ and S22 is nonsingular. That is, S ∈ SΠ . ′ ′ S21 S22 [ ] S11 S12 Proof. Let S = ∈ Sn and, with the notations defined before theorem 2.10, S21 S22 Rα (S22 ) ∈ MI(S22 ) where α ⊂ [n]. By Theorem 2.10, the rows of Rα (S22 ) and Rα′ (S12 ) constitute a nonsingular n × n matrix. Now choose v ∈ {0, 1}n such that vj = 1 for j ∈ α′ . Then the swap matrix Πv ∈ Pn will move the j-th row to the (n + j)-th row and vice versa. To make it clear, let [ ′ ] [ ][ ] ′ S11 S12 diag(b v ) diag(v) S11 S12 Πv S = = . ′ ′ S21 S22 −diag(v) diag(b v) S21 S22 ′ By the definition of swap matrix, we see that Rj (S22 ) = −Rj (S12 ) for j ∈ α′ and ′ ′ constitute a nonsingular n × n ) = Rj (S22 ) for j ∈ α. Hence the rows of S22 Rj (S22 matrix.. Theorem 3.3 (Classification of Symplectic Matrices). (i) For each S ∈ Sn , there exists a swap matrix Π ∈ Pn and a Hermitian matrix ] [ X11 X12 ∈ C2n×2n X= ∗ X22 X12 such that. [ ΠS =. (ii) Sn =. ∪ Π∈Pn. I X11 ∗ O X12. ]−1 [. X12 O X22 I. ] .. SΠ .. (iii) For each Π ∈ Pn , there exists S ∈ SΠ such that S ∈ / SΠ′ for all Π′ ̸= Π. (iv) For each Π ∈ Pn , SΠ is an open set relative to Sn . ] [ S11 S12 ∈ Sn . By Lemma 3.2, there Proof. We first prove assertion (i). Let S = S21 [ S′ 22 ′ ] S11 S12 ′ and S22 is nonsingular. exists a swap matrix Π ∈ Pn such that ΠS = ′ ′ S22 S21 ] [ X11 X12 such that Then by Lemma 3.1, there exists a Hermitian matrix X = ∗ X22 X12 [ ΠS =. I X11 ∗ O X12. ]−1 [. 16. X12 O X22 I. ] ..

(27) Assertion (i) follows. Now we prove assertion (ii). By Lemma 3.2,∪for every S ∈ Sn , there is a swap matrix Π ∈ Pn such that S ∈ SΠ . Hence S ⊂ Π∈Pn SΠ . On the other hand,∪by Definition 1.4 of SΠ , every element of ∪SΠ is an n × n symplectic matrix. Therefore Π∈Pn SΠ ⊂ Sn . We conclude that Sn = Π∈Pn SΠ . Next we prove assertion (iii). Let Π ∈ Pn . By Definition 1.3 there exsits a corresponding vector v ∈ {0, 1}n such that [ ] diag(b v ) diag(v) Π = Πv = . −diag(v) diag(b v) By some computation and Lemma 2.6 we have [ ][ ] diag(b v ) diag(v) diag(b v ) diag(v) Πv Πv = −diag(v) diag(b v) −diag(v) diag(b v) [ ] diag(b v )diag(b v ) − diag(v)diag(v) diag(b v )diag(v) + diag(v)diag(b v) = −diag(v)diag(b v ) − diag(b v )diag(v) −diag(v)diag(v) + diag(b v )diag(b v) [ ] diag(b v ) − diag(v) O = . O diag(b v ) − diag(v) Since [diag(b v ) − diag(v)] is nonsingular, we see that Πv ∈ SΠv . To make it clear, give an illustration as following:     1 1 0 0 0 1     1 1 0 0 0 −1     ..   ..   .. ... ... ...  .  .   .   =  1 1 1  0   0  0     −1 0 −1 0 0 −1     .. .. .. .. .. .. . . . . . .. we      .   . On the other hand, let v ′ ∈ {0, 1}n such that v ′ ̸= v. Then there exists p ∈ [n] such that vp = 1 but vp′ = 0 or q ∈ [n] such that vq = 0 but vq′ = 1. In the case of vp = 1 but [ ] S11 S12 ′ ∗ vp = 0, Cn+p (Πv ) = ep and Rn+p (Πv′ ) = en+p . Thus in Πv′ Πv = , Cp (S22 ) S21 S22 is a 0 vector. Hence Πv ∈ / SΠv′ . We illustrate this as following:     .. .. .. .. .. .. . . . . . .         0 1 1 1 0 0     .. ..   .. ..   .. ..  . .  . .   . .  =  .     .. .. ..  ..   ...   ... . . .          1 0 −1 0 0 −1     .. .. .. .. .. .. . . . . . . 17.       .    .

(28) In the case of vq = 0 but vq′ = [ ] S11 S12 Πv′ Πv = , Cq (S22 ) S21 S22 following:   .. .. . .     0 1   ... ...       .  .  ..  ..     −1 0   ... .... 1, Cn+q (Πv ) = en+q and Rn+q (Πv′ ) = −e∗q . Thus, in is a 0 vector. Hence Πv ∈ / SΠv′ . It is illustrated as. ... ... .. . . .. ... .. ...  ..       0 1    ... ...   ... ...     = . . .. .. .   ..  .. . .       0 1 −1 0    ... ... ... ... 1. 0. We have proved that for each class of SΠ , Π is the very element which lies only in SΠ and does not lie in any other class. Hence this classification is a minimal classification of Sn . Since there are 2n distinct vectors in {0, 1}n , there are 2n corresponding swap matrices. So Sn is classified into 2n classes. Finally we are going to prove that each SΠ is an open set in Sn . The metric we adopt here is defined by ( ∥R − S∥ ..=. ∑. )1/2 | Rij − Sij |2. (3.6). 1≤i,j≤2n. for R, S ∈ C2n×2n . To prove SΠ is open, we need to show that for any given S ∈ SΠ , there exists δ > 0 such that R ∈ Sn together with ∥R − S∥ < δ implying that R ∈ SΠ . Let Πv ∈ [Pn be given] and α ⊂ [n] such that vj = [1 for j ∈ α]and vj = 0 for j ∈ / α. ′ ′ S12 S11 S12 S11 ′ where det(S22 ) ̸= 0. Let S = ∈ SΠ be fixed and ΠS = ′ ′ S S S21 S22 21 22 [ ] A11 A12 ′ be any symplectic matrix in Sn and Denote d = det(S22 ). Let A = A21 A22 [ ′ ] A11 A′12 ΠA = . Since determinant itself is a polynomial function, it is continuous. A′21 A′22 Hence there exists some δ > 0 such that d − |d|/2 < det(A′22 ) < d + |d|/2 ′ ′ ∥ < δ, then det(A′22 ) ̸= 0. Since for any ∥ < δ. That is, if ∥A′22 − S22 for ∥A′22 − S22 ′ ′ A ∈ Sn with ∥A − S∥ < δ, ∥A22 − S22 ∥ ≤ ∥A − S∥ < δ, which implies det(A′22 ) ̸= 0. Threfore SΠ is open.. 18.

(29) 4. The Classification of Symplectic Pairs. We have classified symplectic matrices in the previous section. Now we are going to classify regular symplectic pairs. In [8] Mehrmann and Poloni rearranged the blocks of given symplectic pair ingeniously, which makes a connection between regular symplectic pairs and symplectic matrices. By this rearrangement some properties of Lagrangian subspaces and symplectic matrices can be applied to classify symplectic pairs. For succinct statements, we first give the following definitions. ([ ] [ ]) A11 A12 B11 B12 Definition 4.1. Let (A, B) = , ∈ SPn . The matrix A21 A22 B21 B22 S(A,B) ∈ C4n×4n is defined by  ∗  ∗ ∗ ∗ B11 B21 −B12 −B22  A∗12 A∗22 −A∗11 −A∗21   S(A,B) ..=  ∗ ∗ ∗ ∗ .  B12 B22 B11 B21 A∗11 A∗21 A∗12 A∗22 Definition 4.1 is due to the work in [8] mentioned above. In fact, we shall prove that the matrix S(A,B) is symplectic later in Proposition 4.3. From the above definition, we see two 2n × 2n symplectic matrices merging into one 4n × 4n symplectic matrix. To connect the two different dimensional spaces, we need the following definitions. Definition 4.2. Let v1 , v2 ∈ {0, 1}n . v1 ⊕ v2 ∈ {0, 1}2n is defined by { (v1 )j , for 1 ≤ j ≤ n, (v1 ⊕ v2 )j = (v2 )j−n , for n + 1 ≤ j ≤ 2n. Definition 4.3. Let Π1 , Π2 ∈ Pn and v1 , v2 ∈ {0, 1}n such that Π1 = Πv1 and Π2 = Πv2 . Π1 ⊕ Π2 ∈ P2n is defined by Π1 ⊕ Π2 = Πv1 ⊕v2 . Now we prove some lemmas which will be used in our main theorem. Lemma 2.13 shows some connection between an orthogonal basis of a Lagrangian subspace of C2n and a symplectic matrix in C2n×2n . With the help of Lemma 2.13, Lemma 3.2 can be applied in Lemma 4.1 to show that for any Lagrangian subspace basis matrix U , the rows of U can be rearranged to obtain a nonsingular block we need. The proof of Lemma 4.1 is mainly referred to Theorem 3.1 in [8]. Lemma 4.1. If the columns of U ∈ C2n×n[ span ]a Lagrangian subspace of C2n , then YΠ there exists some Π ∈ Pn such that ΠU = , where Y Π , Z Π ∈ Cn×n and Y Π is ZΠ nonsingular. 19.

(30) Proof. Let U = QR be a QR factorization, where Q ∈ C2n×n is an orthogonal matrix and R ∈ Cn×n is an upper triangle matrix. Since the columns of U span a Lagrangian subspace, U is of full column rank and thus R is nonsingular. (If R is singular, then there exists some column vector v ̸= 0 such that Rv = 0. It follows that U v = QRv = 0. This contradicts that U is of full column rank.) Hence Q = U R−1 is also of full column [ ] Q1 rank and its columns span a Lagrangian subspace. Let Q be partitioned as Q = Q2 n×n where Q1 , Q2 ∈ C . Since the columns of Q are orthogonal and span a Lagrangian subspace, by Lemma 2.13, [ ] Q1 −Q2 Q2 Q1 is orthogonal and symplectic. Let Rα (Q1 ) ∈ MI(Q1 ), then by Corollary 2.11, the rows of Rα′ (Q2 ) together with the rows of Rα (Q1 ) constitute an n × n nonsingular matrix. If we pick v ∈ {0, 1}n such that { 0, i ∈ α, vi = 1, i ∈ / α, [ ′ ] Q1 with Q′1 then with the associated swap matrix Πv ∈ Pn , we have that Πv Q = ′ Q 2 [ ′ ] [ ′ ] Q1 Q1 R nonsingular. Hence in Πv U = Πv QR = R= we also have that Q′1 R ′ Q2 Q′2 R is nonsingular. [ ] I Lemma 4.2. Let U = ∈ C2n×n where I, X ∈ Cn×n and I is an identity matrix. X If the columns of U span a Lagrangian subspace, then X is Hermitian. [ ] I Proof. Since the columns of span a Lagrangian subspace, by Proposition 2.12 X we know that I ∗ X = X ∗ I. That is, X = X ∗ . Therefore X is Hermitian. Lemma 4.1 shows that we can rearrange a Lagrangian subspace basis matrix to obtain a nonsingular block. Combining this to Lemma 4.2, it shows that by multiplying the inverse of the nonsingular block, we can obtain a Hermitian matrix in the other block. Now we shall prove that S(A,B) is symplectic we claim after Definition 4.1. This proof mainly refers to Theorem 6.1 in [8]. Proposition 4.3. If (A, B) ∈ SPn , then S(A,B) ∈ C4n×4n is symplectic. That is, S(A,B) ∈ S2n .. 20.

(31) Proof. Since (A, B) is a symplectic pair, (A, B) satisfies the condition AJA∗ = BJB ∗ , which implies that [ ] [ ] ∗ ∗ ∗ ∗ −A12 A∗11 + A11 A∗12 −A12 A∗21 + A11 A∗22 −B12 B11 + B11 B12 −B12 B21 + B11 B22 = . ∗ ∗ ∗ ∗ −A22 A∗11 + A21 A∗12 −A22 A∗21 + A21 A∗22 −B22 B11 + B21 B12 −B22 B21 + B21 B22 (4.1) Equation (4.1) can be rewritten as   ∗  ∗ O O I O B B 11 21 [ ]   ∗ A∗22  B11 A12 B12 A11   O O O I   A12  (4.2) ∗ ∗  = O. B21 A22 B22 A21  −I O O O   B12 B22 O −I O O A∗11 A∗21 [ ] B11 A12 B12 A11 Moreover, we see that the 2n × 4n matrix is of full row rank, B21 A22 B22 A21 [ ]∗ B11 A12 B12 A11 for otherwise we can find a nonzero vector w such that w = 0, B21 A22 B22 A21 [ ] B11 A12 B12 A11 i.e., w∗ = 0. By shuffling the columns we have B21 A22 B22 A21 [ ] [ ] B11 B12 A11 A12 ∗ ∗ w =w = 0. B21 B22 A21 A22 Hence w∗ A − λw∗ B = w∗ (A − λB) = 0 for all λ ∈ C,[contradicting the regularity ] B11 A12 B12 A11 assumption that det(A − λB) ̸= 0 for some λ ∈ C. Since is of B21 A22 B22 A21 ]∗ [ B11 A12 B12 A11 full row rank, is of full column rank. Combining this result B21 A22 B22 A21 [ ]∗ B11 A12 B12 A11 with equation (4.2), we conclude that the columns of span a B21 A22 B22 A21 Lagrangian subspace of C4n . Then by Lemma 2.13,   ∗ ∗ ∗ ∗ −B22 −B12 B11 B21  A∗12 A∗22 −A∗11 −A∗21   S(A,B) =  ∗ ∗ ∗ ∗   B12 B22 B11 B21 A∗11 A∗21 A∗12 A∗22 is a symplectic matrix. Before proving Proposition 4.5, which shows close relationship between (A, B) ∈ SPn and S(A,B) ∈ S2n , we need an easily seen lemma. 21.

(32) [ Lemma 4.4. If ΠS = ′ ′ det (S11 ) = det (S22 ).. ′ ′ S11 S12 ′ ′ S21 S22. ]. [ where S =. S1 −S2 S2 S1. ] ∈ Sn and Π ∈ Pn , then. Proof. Let v ∈ {0, 1}n such that Π = Πv and α ⊂ [n] such that vj = 1 for j ∈ α ′ ′ and vj = 0 for j ∈ / α. Then for j ∈ / α, Rj (S22 ) = Rj (S1 ) = Rj (S11 ), and for j ∈ α, ′ ′ ′ ′ ′ Rj (S22 ) = −Rj (−S2 ) = Rj (S2 ) = Rj (S11 ). That means S22 = S11 . Hence det (S22 )= ′ det (S11 ). The close relationship between a regular symplectic pair (A, B) and its corresponding symplectic matrix S(A,B) is more than Proposition 4.3 tells. The following proposition gives us an insight to see why we can classify regular symplectic pairs with some results obtained from symplectic matrices. Proposition 4.5. Let (A, B) ∈ SPn and Π1 , Π2 ∈ Pn . Then (A, B) ∈ SPΠ2 ,Π1 if and only if S(A,B) ∈ SΠ1 ⊕Π2 . Proof. Let v1 , v2 ∈ {0, 1}n such that Π1 = Πv1 and Π2  ∗  B11 diag(v1 ) O diag(b v1 ) O ∗    O diag(v ) O diag(b v ) 2   A12 2  ∗   B12  −diag(v1 ) O diag(b v1 ) O ∗ O −diag(v2 ) O diag(b v2 ) A11. = Πv2 . If S(A,B) ∈ SΠ1 ⊕Π2 , then  ∗ ∗ ∗ B21 −B22 −B12 [ ′ ] ′ S11 S12 A∗22 −A∗11 −A∗21   ∗ ∗ ∗  = ′ ′ B22 B11 B21 S21 S22 ∗ ∗ ∗ A21 A22 A12 (4.3) ′ ′ and S22 is nonsingular. By Lemma 4.4 S[11 is also nonsingular. ]Moreover, by Propo∗ B11 A12 B12 A11 sition 2.14, we see that the columns of span a Lagrangian B21 A22 B22 A21 subspace. From equation (4.3) we can derive that   ∗ ∗ B11 B21 [ ′ ] [ ]  A∗12 A∗22  I S ′ 11  = S11 (4.4) (Π1 ⊕ Π2 )  ′ ∗ ∗  =  B12 B22 S21 X A∗11 A∗21 ′−1 ′ where X = S21 S11 . Since the swap matrix [ ]Π1 ⊕ Π2 does not change the structure I of Lagrangian subspace, the columns of also spans a Lagrangian subspace. By X [ ] X11 X12 Lemma 4.2 we see that X is Hermitian. Let X = . Then we have X21 X22 ∗ ∗ ∗ X11 = X11 , X12 = X21 , and X22 = X22 .. 22. (4.5).

(33) Rewrite equation (4.4) we have [ ] ][ ∗ ] [ ∗ diag(b v1 ) diag(v1 ) B11 B21 I O ′ = S11 ∗ ∗ −diag(v1 ) diag(b v1 ) B12 B22 X11 X12 [. (4.6). ][. ] [ ] A∗12 A∗22 O I ′ . (4.7) = S11 A∗11 A∗21 X21 X22 [ ] I O ∗ ′ Equation (4.6) can be written as Π1 B = S11 . Take the conjugate transX11 X12 pose of both sides and by (4.5) we have [ [ ] ] ∗ I X11 I X11 ∗ ′∗ ′∗ . (4.8) BΠ1 = S11 = S11 ∗ ∗ O X12 O X12 and. diag(b v2 ) diag(v2 ) −diag(v2 ) diag(b v2 ). ∗ By Proposition 2.7 we know that Π−1 1 = Π1 . Hence we have [ ] I X11 ′∗ B = S11 Π1 . O X21. (4.9). Now we turn to equation (4.7). Take the conjugate transpose of both sides we have ] [ ] [ ∗ A12 A11 O X21 ∗ ′∗ Π2 = S11 ∗ A22 A21 I X22 [ ] O X12 ′∗ = S11 . (4.10) I X22 Rewrite the left side of (4.10) as following [ ] [ ][ ][ ] A12 A11 A12 A11 O I O I ∗ Π2 = Π∗2 A22 A21 A22 A21 I O I O ][ ] [ A11 A12 O I = Π∗2 . A21 A22 I O. 23. (4.11).

(34) Combining (4.10) and (4.11) and moving some terms from left to right we then have [ ] [ ] O X12 O I ′∗ Π2 A = S11 I X22 I O [ ][ ] O X12 diag(v2 ) diag(b v2 ) ′∗ = S11 I X22 diag(b v2 ) −diag(v2 ) ][ ][ ][ ] [ O X12 O I O I diag(v2 ) diag(b v2 ) ′∗ = S11 I X22 I O I O diag(b v2 ) −diag(v2 ) [ ][ ] X12 O diag(b v2 ) −diag(v2 ) ′∗ = S11 X22 I diag(v2 ) diag(b v2 ) [ ] X12 O ′∗ (4.12) Π∗2 . = S11 X22 I By (4.9), (4.12), and Definition 1.2, we ([ X12 (A, B) ∼ X22 [ X11 for some Hermitian matrix X = ∗ X12 if (A, B) ∈ SPΠ2 ,Π1 , then ([ X12 (A, B) ∼ X22 [ X11 for some Hermitian matrix X = X21. conclude that ] [ ] ) O I X11 ∗ Π2 , Π1 ∗ I O X12 ] X12 . That is, (A, B) ∈ SPΠ2 ,Π1 . Conversely, X22 O I. ]. [ Π∗2 ,. I X11 O X21. ] X12 . That is, X22 [ ] I X11 B=M Π1 O X21 [. and A=M. X12 O X22 I. ]. ) Π1. (4.13). ] Π∗2. (4.14). for some nonsingular matrix M . From equations (4.6), (4.8), and (4.9) we see that equation (4.13) can be rewritten as [ ][ ∗ ] [ ] ∗ diag(b v1 ) diag(v1 ) B11 B21 I O = M ∗. (4.15) ∗ ∗ −diag(v1 ) diag(b v1 ) B12 B22 X11 X12 24.

(35) From equations (4.7), (4.10), (4.11), and (4.12) we see that equation (4.14) can be rewritten as [ ] ][ ∗ ] [ diag(b v2 ) diag(v2 ) A12 A∗22 O I = M ∗. (4.16) −diag(v2 ) diag(b v2 ) A∗11 A∗21 X21 X22 Combining equations (4.15) and (4.16) and rearranging  ∗  diag(v1 ) O B11 diag(b v1 ) O ∗    O diag(b v ) O diag(v ) 2 2   A12  ∗   B12  −diag(v1 ) O diag(b v1 ) O O −diag(v2 ) O diag(b v2 ) A∗11. the blocks we obtain   ∗ B21 I O  O A∗22  I   ∗  =  X11 X12 B22 ∗ X21 X22 A21.   ∗ M .  (4.17). That is,. . ∗ B11  A∗12 (Π1 ⊕ Π2 )  ∗  B12 A∗11.  ∗ B21 [ ] M∗ A∗22   . ∗  = B22 XM ∗ A∗21. Suppose that. [ (Π1 ⊕ Π2 )S(A,B) = [. ′ ′ S12 S11 ′ ′ S21 S22. (4.18). ] .. (4.19). ] S1 −S2 ′ ′ ) = ) = det (S11 Since S(A,B) is of the form , by Lemma 4.4, det (S22 S2 S1 det (M ∗ ) ̸= 0. Thus S(A,B) ∈ SΠ1 ⊕Π2 . Now we proceed to prove the main theorem of this section. Theorem 4.6 (Classification of Regular Symplectic Pairs). (i) For each regular symplectic pair (A, B) ∈ SPn , there exist swap matrices Π1 , Π2 ∈ Pn and a Hermitian matrix [ ] X11 X12 ∈ C2n×2n X= ∗ X12 X22 ([. such that (A, B) ∼. X12 O X22 I. ]. [ Π∗1 ,. That is, (A, B) ∈ SPΠ1 ,Π2 . ∪ (ii) SPn = Π1 ,Π2 ∈Pn SPΠ1 ,Π2 . (iii) Each SΠ1 ,Π2 is an open set relative to SPn . 25. I X11 ∗ O X12. ]. ) Π2 ..

(36) Proof. We first prove assertion (i). This proof mainly refers to [8]. Let (A, B) ∈ SPn . By Proposition 4.3, S(A,B) ∈ S2n . Then by Lemma 3.2, there exists Π ∈ P2n such that S(A,B) ∈ SΠ . Let Π1 , Π2 ∈ Pn such that Π1 ⊕ Π2 = Π. By Proposition 4.5, (A, B) ∈ SPΠ2 ,Π1 . That is, [ ] ) ([ ] I X11 X12 O ∗ Π1 . (A, B) ∼ Π2 , ∗ O X12 X22 I [ ] X11 X12 for some Hermitian matrix X = ∈ C2n×2n . Assertion (ii) is simply a ∗ X12 X22 corollary of assertion (i). Since for each regular symplectic ∪ pair (A, B), there eixst Π1 , Π2 ∈ Pn such that (A, B) ∈ SPΠ1 ,Π2 , we have SPn ⊂ Π1 ,Π2 ∈Pn SPΠ1 ,Π2 . On the other hand, by Definition 1.4 we know that each element of SPΠ1 ,Π2 is an element of ∪ SPn . Hence SPn = Π1 ,Π2 ∈Pn SPΠ1 ,Π2 . Since there are 22n elements in Pn × Pn , we break SPn into 22n classes. However, this classification is not minimal. We will give an example later. Finally we are going to prove that each SPΠ1 ,Π2 is an open set in SPn . The metric we adopt here is defined by ( )1/2 ∑ ∥(A1 , B1 ) − (A2 , B2 )∥ ..= | (A1 )ij − (A2 )ij |2 + | (B1 )ij − (B2 )ij |2 (4.20) 1≤i,j≤2n. for (A1 , B1 ), (A2 , B2 ) ∈ SPn . Let Π1 , Π2 ∈ Pn be fixed and assume that (A, B) ∈ SΠ1 ,Π2 . By Proposition 4.5, S(A,B) ∈ SΠ2 ⊕Π1 ⊂ S2n . Then by Theorem 3.3 (iv), SΠ2 ⊕Π1 is relatively open to S2n . That is, there exists some δ > 0 such that S ∈ S2n together with ∥S − S(A,B) ∥ < δ implying that S ∈ SΠ2 ⊕Π1 . By the definitions of metric we adopted (see equations (3.6) and (4.20)) and Definition 4.1 of S(A,B) , we see that ∥S(C,D) − S(A,B) ∥2 = 2∥(C, D) − (A, B)∥2. (4.21). √ for any (C, D) ∈ SPn with S(C,D) ∈ S2n . If we choose d = δ/ 2. Then ∥(C, D) − (A, B)∥ < d implies that ∥S(C,D) − S(A,B) ∥ < δ, and thus S(C,D) ∈ SΠ2 ⊕Π1 . Applying Proposition 4.5 again, we conclude that (C, D) ∈ SΠ1 ,Π2 . Comparing Theorems 3.3 and 4.6, we can see in Theorem 4.6 there is no correspondent statement to Theorem 3.3 (iii), “for each SPΠ1 ,Π2 , there exists (A, B) ∈ SPΠ1 ,Π2 such that (A, B) ∈ / SΠ′1 ,Π′2 for all Π′1 ̸= Π1 or Π′2 ̸= Π2 .” In fact, this statement could not be true in P1 . We give an example as following. [ ] [ ] 1 0 0 1 Example 4.1. Let I = and J = . Then SPI,J ⊂ SP1 is such a 0 1 −1 0 class that each element in SPI,J must also be in another class. 26.

(37) Proof. There are only four classes in SP1 , i.e., SPI,I , SPI,J , SPJ,I , and SPJ,J . Suppose that (A, B) ∈ SP1 such that (A, B) ∈ SPI,J but (A, B) ∈ / SPI,I , (A, B) ∈ / SPJ,I , and (A, B) ∈ / SPJ,J . By Proposition 4.5, we have that S(A,B) ∈ SI⊕J , S(A,B) ∈ / SI⊕I , S(A,B) ∈ / SJ⊕I , and S(A,B) ∈ / SJ⊕J . (4.22) [ ] a b For convenience, we just consider real entries here. Let A = and B = c d   f h −g −k [ ]  b d −a −c  f g  . Doing some computatione we have . Then S(A,B) =   g k h k f h  a c b d that   ∗ ∗ (4.23) ΠI⊕J S(A,B) =  f h , ∗ a c   ∗ ∗ (4.24) ΠI⊕I S(A,B) =  f h , ∗ b d   ∗ ∗ ΠJ⊕I S(A,B) =  (4.25) g k  , and ∗ b d   ∗ ∗ ΠJ⊕J S(A,B) =  (4.26) g k . ∗ a c By equations (4.22) to (4.26) and Definition 1.4 of SΠ , we have that

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(39)

(40)

(41)

(42)

(43)

(44)

(45)

(46)

(47) f h

(48)

(49)

(50)

(51)

(52)

(53)

(54)

(55) ̸= 0 and

(56) f h

(57) =

(58) g k

(59) =

(60) g k

(61) = 0. (4.27)

(62) a c

(63)

(64) b d

(65)

(66) b d

(67)

(68) a c

(69) [ ] [ ] [ ] [ ] f a g b Hence ̸= 0 and ̸= 0. As to and , we discuss in four cases. h c k d [ ] [ ] g b (Case 1) If = = 0, then k d [ ] [ ] a b f g A − λB = −λ c d h k [ ] a − λf 0 = . c − λh 0 27.

(70) This would lead a contradiction to the regularity assumption. [ ] [ ] [ ] g b tf (Case 2) If = 0 and = ̸= 0, compute k d th [ ][ ][ ] f g 0 1 f h ∗ BJB = h k −1 0 g k [ ][ ][ ] f 0 0 1 f h = h 0 −1 0 0 0 [ ][ ] 0 f f h = 0 h 0 0 [ ] 0 0 = 0 0 and. [. ][ ] 0 1 a c AJA = −1 0 b d [ ][ ][ ] a tf 0 1 a c = c th −1 0 tf th [ ][ ] −tf a a c = −th c tf th [ ] 0 −tf c + ath = . −tha + ctf 0 ∗. a b c d. ][. Since (A, B) is a symplectic

(71) pair, we see that ah = cf . But this contradicts

(72)

(73) f h

(74)

(75) ̸= 0. equation (4.27) that

(76)

(77) a c

(78) [ ] [ ] [ ] b g ra (Case 3) If = 0 and = ̸= 0, this case is similar to Case 2. d k rc [ ] [ [ ] [ ] ] b tf g ra (Case 4) If = ̸= 0 and = ̸= 0, then d th k rc

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(80)

(81)

(82)

(83)

(84)

(85) g k

(86)

(87) ra rc

(88)

(89)

(90)

(91)

(92) =

(93)

(94) = rt

(95) a c

(96) ,

(97) b d

(98)

(99) tf th

(100)

(101) f h

(102) contradicting equation (4.27). So we conclude that such (A, B) does not exist and that each element in SPI⊕J must be also in some other class. 28.

(103) 5. Conclusion and Future Works. In this thesis we classify symplectic matrices and regular symplectic pairs. The main idea is to “swap” the rows of a given symplectic matrix to obtain a nonsingular block, and thus classify symplectic matrices with such “swap matrices”. As to regular symplectic pairs, we rearrange and merge them into symplectic matrices. Then classify them with similar thought. But Example 4.1 shows that the classifications we obtain on regular symplectic pairs are not minimal. So we may consider some other way to find a minimal classification of regular symplectic pairs. Here is one simple way to reduce the classes of symplectic pairs we obtained. If (A, B) ∈ SPn with B nonsingular, then AB −1 is symplectic. By Theorem 1.1, [ ΠAB. −1. = [. for some Hermitian matrix X =. I X11 ∗ O X12. X11 X12 ∗ X22 X12. ]−1 [. −1. ] .. ] . Then we have ([. −1. X12 O X22 I. (A, B) ∼ (AB , I) ∼ (ΠAB , Π) ∼. X12 O X22 I. ] [ ] ) I X11 , Π . ∗ O X12. That is, (A, B) ∈ SPI,Π . Hence for B nonsingular, the number of classes of (A, B) can be reduced to only 2n , far less than 22n . As to both A and B nonsigular, there are still problems to be solved. This minimal classification problem can be our later study.. References [1] R. Abraham and J. Marsden. Foundations of Mechanics, second ed. AddisonWesley, Reading, 1978. [2] V.I. Arnold. Mathematical Methods in Classical Mechanics. Berlin, 1978.. Springer-Verlag,. [3] Froiln M. Dopico and Charles R. Johnson. Complementary bases in symplectic matrices and a proof that their determinant is one. Linear Algebra and its Applications, 419(2):772 – 778, 2006. [4] H. Fassbender. Symplectic Methods for the Symplectic Eigenproblem. SpringerVerlag, Berlin, 2000.. 29.

(104) [5] D.S. Mackey and N. Mackey. On the Determinant of Symplectic Matrices, Numerical Analysis Report No. 422. Manchester Centre for Computational Mathematics, Manchester, England, 2003. [6] D. McDuff and D. Salamon. Introduction to Symplectic Topology. Clarendon Press, Oxford, 1995. [7] V. Mehrmann. The autonomous linear quadratic control problem - theory and numerical solutions, lecture notes in control and information sciences, vol. 163, 1991. [8] Volker Mehrmann and Federico Poloni. Doubling algorithms with permuted lagrangian graph bases. SIAM Journal on Matrix Analysis and Applications, 33(3):780–805, 2012.. 30.

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