An Elementary Proof That Symplectic Matrices Have Determinant One 3

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2. A proof that the determinant of a real or complex symplectic matrix is one Let us denote the set of 2N 2N symplectic matrices over field K by Sp(2N, K). A feature of the proofs below× is that they follow a similar overall strategy for both cases, when K = R and K = C. Nonetheless, in detail there is nontrivial difference arising from field-dependence, in the form of Lemma 2.2. We first give the proof when A is real. Theorem 2.1. Let A Sp(2N, R). Then det(A) = 1. ∈ Proof. Taking the determinant on both sides of AT JA = J, T T (2.1) det(A JA) = det(A ) det(J) det(A) = det(J). So we immediately have that det(A)= 1. Then let us consider the matrix ± T (2.2) A A + I. Since AT A is symmetric positive definite, the eigenvalues of (2.2) are real and greater than 1. Therefore its determinant, being the product of its eigenvalues, has det(AT A + I) > 1. Now as det(A) = 0, A is invertible. Using (1.1) we may write 6 T T T T (2.3) A A + I = A (A + A− )= A (A + JAJ −1). Denote the four N N subblocks of A as follows, × A11 A12 N×N (2.4) A = , where A11, A12, A21, A22 R . A21 A22 ∈ Then we compute (2.5) −1 A11 A12 O IN A11 A12 O IN A + JAJ = + − A21 A22 IN O A21 A22IN O − A11 A12 A22 A21 A11 + A22 A12 A21 (2.6) = + − = − . A21 A22 A12 A11 A12 + A21 A11 + A22 − − Writing the blocks as C := A11 +A22 and D := A12 A21, we make use of a unitary transform − CD A + JAJ −1 = D C − 1 IN IN C + iD O 1 IN iIN (2.7) = − . √2 iIN iIN O C iD √2 IN iIN − − We plug this factorization into (2.3). Note that C,D are both real. This allows the complex conjugation to commute with the determinant (as it is a polynomial of its entries) T T 0 < 1 < det(A A + I) = det(A (A + JAJ −1)) = det(A) det(C + iD) det(C iD) − = det(A) det(C + iD) det C + iD 2 = det(A) det(C + iD)det(C + iD) = det(A) det(C + iD) . | | AN ELEMENTARY PROOF THAT SYMPLECTIC MATRICES HAVE DETERMINANT ONE 3

Clearly, none of the two determinants on the RHS can be zero, so we may conclude 2 det(C + iD) > 0. Dividing this through on both sides, we have det(A) > 0, and |thus det(A) =| 1. The unitary transform (2.7) appears in the final part of a proof that the determinant of any real symplectic matrix is +1 using polar decomposition in [3] although there it is not used in conjunction with (2.2). In our proof its use was in showing that CD N×N (2.8) det 0 forany C,D R . D C ≥ ∈ − The next lemma is a generalization of (2.8) to complex matrices, and is the main hurdle in applying the above approach to the complex field. It arises naturally in the treatment of quaternionic matrices, and was proved in [8, 9]. We simply reproduce the latter proof here for completeness. CD Lemma 2.2. Let C,D CN×N . Then det 0. ∈ D C ≥ − Proof. Let us first assume that C is invertible. Then letting E := C−1D, det EE + I 0 since eigenvalues of EE occur in ≥ conjugate pairs and every negative eigenvalue of EE has even algebraic multiplicity [10]. Note that E and I commute, which implies that I E det 0. E I ≥ − Then we have, − C 1 0 CD I C−1D I E −1 = = . 0 C D C C−1D I E I − − − Taking the determinant on both sides, we have the result. The Lemma follows by continuity. Now we prove the complex case. Given Lemma 2.2 the proof is very similar to the real case. Theorem 2.3. Let A Sp(2N, C). Then det(A) = 1. ∈ Proof. If A is symplectic then we again have from (2.1) that det(A)= 1. Consider the matrix ± (2.9) A∗A + I. The matrix A∗A is Hermitian positive definite and therefore the matrix (2.9) has eigenvalues greater than one, and thus det(A∗A + I) > 1. Then we factor, (2.10) A∗A + I = A∗(A + A−∗)= A∗(A + A−T ). Then we again use the definition (1.1) and with the notation (2.4) for subblocks,

A11 A12 A22 A21 A + JAJ −1 = + − A21 A22 A12 A11 − A11 + A22 A12 A21 CD = − = , A12 + A21 A11 + A22 D C − − 4 DONSUB RIM if we let C := A11 + A22 and D := A12 A21. Therefore we have from (2.10) − CD CD 0 < 1 < det(A∗(A + A−∗)) = det(A) det = det(A) det . D C D C − − CD By Lemma 2.2 we necessarily have that det > 0, so we divide and D C conclude det(A) > 0, showing that det(A) = 1. −

3. An explicit formula for determinant of conjugate symplectic matrices Conjugate symplectic matrices A C2N×2N are defined similarly to symplectic matrices, by replacing the transpose∈ in (1.1) with the conjugate transpose, A∗JA = J. Again, it immediately follows that det(A) = 1, however det(A) may take on any complex value on the unit circle [5].| Turning| our attention to (2.9) once more, A∗A + I = A∗(A + A−∗)= A∗(A + JAJ −1). With the previous notation for the N N subblocks of A and with C,D defined as × A11 A12 A = , C := A11 + A22,D := A12 A21, A21 A22 − we obtain (2.7) but for a complex matrix. Then it follows that CD 1 < det(A) det = det(A) det(C + iD) det(C iD) D C − − = det(A)det(C2 + D2 i[C,D]), − where [C,D] := CD DC is the commutator. This determines the− phase and therefore det(A), yielding the formula 2 2 det((A11 + A22) + (A12 A21) i[A11 + A22, A12 A21]) (3.1) det(A)= 2 − 2 − − . det((A11 + A22) + (A12 A21) i[A11 + A22, A12 A21]) | − − − | 4. Acknowledgements The author would like to express thanks to Prof. Froilán M. Dopico and Prof. Ioana Dumitriu for carefully reading this paper and providing valuable comments, and thanks Prof. Bernard Deconinck for introducing him to this problem. He also thanks Dr. Meiyue Shao for bringing to attention the reference [1].

References 1. Y. Xu, Linear Algebra and Matrix Theory, (Chinese) 2nd Edition, Higher Education Press, Beijing, China, 2008. 2. F. M. Dopico, C. R. Johnson, Complementary bases in symplectic matrices and a proof that their determinant is one, Linear Algebra Appl. 419 (2006), 772–778. 3. K.L. Meyer, G.R. Hall. Introduction to Hamiltonian Dynamical Systems and the N-Body Problem, Volume 90 of Applied Mathematical Sciences, Springer, New York., 1991. 4. E. Artin. Geometric Algebra, Interscience Tracts in Pure and Applied Mathematics. Inter- science Publishers Inc., New York, 1957. 5. D.S. Mackey, N. Mackey, On the determinant of symplectic matrices, Numerical Analysis Report No. 422, Manchester Centre for Computational Mathematics, Manchester, England, (2003). AN ELEMENTARY PROOF THAT SYMPLECTIC MATRICES HAVE DETERMINANT ONE 5

6. D.S. Mackey, N. Mackey, F. Tisseur, Structured tools for structured matrices, Electron. J. Linear Algebra 419 (2003), 106–145. 7. D.S. Mackey, N. Mackey, F. Tisseur, G-reﬂectors in scalar product spaces. Linear Algebra Appl. 385 (2004), 187–213. 8. H. Aslaksen, Quaternionic Determinants, Math. Intell. 18(3) (1996), 57–65. 9. F. Zhang, Quaternions and matrices of quaternions, Linear Algebra Appl. 251 (1997), 21–57. 10. R. A. Horn and C.R. Johnson, Matrix Analysis, Cambridge University Press, 1990.

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