ON SOME PROPERTIES OF THE SYMPLECTIC AND
HAMILTONIAN MATRICES
by
Dorin Wainberg
Abstract: In the first part of the paper the symplectic and Hamiltonian matrices are defined and some properties are pointed, and in the second part a relation between those two sets of matrices is proved.
1. Proprieties of symplectic and Hamiltonian matrices
For the beginning it will be introduced the square matrix J ∈ M2n×2n() defined by
J =
− n n
n n
I I
0 0
(1)
where: 0n∈ Mn×n() – zero matrix In∈ Mn×n() – identity matrix Remark 1
It is not very complicated to prove that the next properties of the matrix J
are real (properties that will be very useful to prove some propositions that follow):
i) Jt = -J
ii) J-1 = Jt iii) JtJ = I2n iv) Jt Jt= - I2n v) J2 = - I2n vi) det J = ±1
Now the definition of the symplectic and Hamiltonian matrices are given:
Definition 1
A matrix A∈ M2n×2n() is called symplectic if:
where J∈ M2n×2n() is from (1). We will denote by SP(n, )
not
= { A ∈M2n×2n() | At J A = J } the set of 2n
× 2n real symplectic matrices. Definition 2
In the next part some properties of those sets of matrices will be proved, for example:
= Jt (J A)tJ J At J = J2 = - I2n = Jt A Jt (- I2n) At J =
= - Jt A Jt At J = A Jt At = (At J A)t = Jt = - Jt Jt J = Jt Jt = - I2n
= J ⇒ A-1 ∈ SP(n, )
We will proof now that AB∈ SP(n, ):
(AB)tJ AB = Bt At J A B = At J A = J
= Bt J B =
= J ⇒ AB∈ SP(n, )
Consequence. The set of symplectic matrices SP(n, ) is a subgroup of the set of nonsingular matrices GL(n, ) reported to multiplication. Indeed we have AB∈ SP(n, ), ∀ A, B∈ SP(n, ), and AB-1∈ SP(n, ), ∀ A, B∈ SP(n, ).
Proposition 2
Let A ∈ SP(n, ) and pA(x) - the characteristic polynomial of the matrix
A. If pA(c)= 0, then pA(1c)= pA(c)= pA(1c)= 0, where c∈ . Proof
) (x
pA = det(A – xI2n) = A = - J A-1J
= det(- J A-1J – xI2n) = I2n = - J2
= det( J( - A-1)J + x J2) = = det( J( - A-1 + xI2n)J ) =
= det J det( - A-1 + xI2n)det J = det J = ±1 = det( - A-1 + xI2n) =
= det( - A-1 + xAA-1) =
= det( - I2n + xA ) det( A-1 )= = ±det(xA - I2n ) = ±x2n det(A - x1
n
I2 ) = ±x2npA(1x) ⇒
⇒ x2npA(1x) = 0 ⇒ pA(1x) = 0 ⇒ pA(1c) = 0 ⇒ pA(c) = 0 ⇒ pA(1c) = 0. Proposition 3
The next relations are equivalent:
a) A is a Hamiltonian matrix; b) A = JS , where S = St ; c) ( JA )t = JA.
i) [ αA + βB, C] = α[A, C] + β[B, C] - evidently ⇒ the operation is bilinear.
ii) [A, B] = AB – BA = - (BA – AB) = - [B, A] ⇒ the operation is antisymmetric.
iii) Jacobi’s relation is satisfied:
[[A, B], C] + [[C, A], B] + [[B, C], A] = [AB – BA, C] + [CA – AC, B] + [BC –CB, A] = =ABC – BAC – (CAB – CBA) + CAB –ACB – (BCA –
BAC) + BCA – CBA – (ABC – ACB) = 0 Proposition 5
Let A ∈ sp(n, ) and pA(x) - the characteristic polynomial of the matrix A. Then:
a) pA(x) = pA(−x)
b) if pA(c)= 0, then pA(−c) = pA(c) = pA(−c) = 0, where c∈ . Proof
a) pA(x)= det(A – xI2n) , but A = J At J ⇒ )
(x
pA = det(J At J – xI2n) A = - J A-1J
= det(J At J – J x J) = = det( J( At + xI2n)J ) =
= det J det(At + xI2n)det J = det J = ±1 = det(At + xI2n) = det(At + xI2nt) =
= det(A + xI2n)t =
= det(A + xI2n) = det(A – (–x )I2n) = pA(−x) b) pA(c)= 0
) a
⇒ pA(−c) = 0. )
(x
pA is a real coefficients polynomial ⇒ pA(c) = 0 ) a
⇒ pA(−c) = 0.
2.A relation between the sets sp(n, ) and SP(n, )
Theorem:
Let A∈ M2n×2n(). The next relations are equivalent: a. A∈ sp (n, )
b. exp (At) ∈ SP (n, ) Proof
„a ⇒ b”
Let U(t) = (exp(At))tJ exp(At)
[1] Mircea Puta – Calcul matriceal, Timişoara, Editura Mirton, 2000
[2] Mircea Puta – Varietăţi diferenţiabile - probleme, Timişoara, Editura Mirton, 2002
[3] Mircea Puta – Hamiltonian Mechanical Systems and Geometric
Quantinization, Kluwer, 1993.
[4] R. Abraham, J. Marsden and T. Ratiu - Manifolds, Tensor Analysis and
Applications, Second Edition, Applied Math. Sciences 75, Springer Verlag,
