Menu
Home Explore People Places Arts History Plants & Animals Science Life & Culture Technology
On this page
Hamiltonian matrix
Mathematical matrix

In mathematics, a Hamiltonian matrix is a 2n-by-2n matrix A such that JA is symmetric, where J is the skew-symmetric matrix

J = [ 0 n I n − I n 0 n ] {\displaystyle J={\begin{bmatrix}0_{n}&I_{n}\\-I_{n}&0_{n}\\\end{bmatrix}}}

and In is the n-by-n identity matrix. In other words, A is Hamiltonian if and only if (JA)T = JA where ()T denotes the transpose. (Not to be confused with Hamiltonian (quantum mechanics))

We don't have any images related to Hamiltonian matrix yet.
We don't have any YouTube videos related to Hamiltonian matrix yet.
We don't have any PDF documents related to Hamiltonian matrix yet.
We don't have any Books related to Hamiltonian matrix yet.
We don't have any archived web articles related to Hamiltonian matrix yet.

Properties

Suppose that the 2n-by-2n matrix A is written as the block matrix

A = [ a b c d ] {\displaystyle A={\begin{bmatrix}a&b\\c&d\end{bmatrix}}}

where a, b, c, and d are n-by-n matrices. Then the condition that A be Hamiltonian is equivalent to requiring that the matrices b and c are symmetric, and that a + dT = 0.23 Another equivalent condition is that A is of the form A = JS with S symmetric.4: 34 

It follows easily from the definition that the transpose of a Hamiltonian matrix is Hamiltonian. Furthermore, the sum (and any linear combination) of two Hamiltonian matrices is again Hamiltonian, as is their commutator. It follows that the space of all Hamiltonian matrices is a Lie algebra, denoted sp(2n). The dimension of sp(2n) is 2n2 + n. The corresponding Lie group is the symplectic group Sp(2n). This group consists of the symplectic matrices, those matrices A which satisfy ATJA = J. Thus, the matrix exponential of a Hamiltonian matrix is symplectic. However the logarithm of a symplectic matrix is not necessarily Hamiltonian because the exponential map from the Lie algebra to the group is not surjective.5: 34–36 6

The characteristic polynomial of a real Hamiltonian matrix is even. Thus, if a Hamiltonian matrix has λ as an eigenvalue, then −λ, λ* and −λ* are also eigenvalues.7: 45  It follows that the trace of a Hamiltonian matrix is zero.

The square of a Hamiltonian matrix is skew-Hamiltonian (a matrix A is skew-Hamiltonian if (JA)T = −JA). Conversely, every skew-Hamiltonian matrix arises as the square of a Hamiltonian matrix.8

Extension to complex matrices

As for symplectic matrices, the definition for Hamiltonian matrices can be extended to complex matrices in two ways. One possibility is to say that a matrix A is Hamiltonian if (JA)T = JA, as above.910 Another possibility is to use the condition (JA)* = JA where the superscript asterisk ((⋅)*) denotes the conjugate transpose.11

Hamiltonian operators

Let V be a vector space, equipped with a symplectic form Ω. A linear map A : V ↦ V {\displaystyle A:\;V\mapsto V} is called a Hamiltonian operator with respect to Ω if the form x , y ↦ Ω ( A ( x ) , y ) {\displaystyle x,y\mapsto \Omega (A(x),y)} is symmetric. Equivalently, it should satisfy

Ω ( A ( x ) , y ) = − Ω ( x , A ( y ) ) {\displaystyle \Omega (A(x),y)=-\Omega (x,A(y))}

Choose a basis e1, …, e2n in V, such that Ω is written as ∑ i e i ∧ e n + i {\textstyle \sum _{i}e_{i}\wedge e_{n+i}} . A linear operator is Hamiltonian with respect to Ω if and only if its matrix in this basis is Hamiltonian.12

References

  1. Ikramov, Khakim D. (2001), "Hamiltonian square roots of skew-Hamiltonian matrices revisited", Linear Algebra and its Applications, 325: 101–107, doi:10.1016/S0024-3795(00)00304-9. /wiki/Doi_(identifier) ↩

  2. Ikramov, Khakim D. (2001), "Hamiltonian square roots of skew-Hamiltonian matrices revisited", Linear Algebra and its Applications, 325: 101–107, doi:10.1016/S0024-3795(00)00304-9. /wiki/Doi_(identifier) ↩

  3. Meyer, K. R.; Hall, G. R. (1991), Introduction to Hamiltonian dynamical systems and the N-body problem, Springer, ISBN 0-387-97637-X. 0-387-97637-X ↩

  4. Meyer, K. R.; Hall, G. R. (1991), Introduction to Hamiltonian dynamical systems and the N-body problem, Springer, ISBN 0-387-97637-X. 0-387-97637-X ↩

  5. Meyer, K. R.; Hall, G. R. (1991), Introduction to Hamiltonian dynamical systems and the N-body problem, Springer, ISBN 0-387-97637-X. 0-387-97637-X ↩

  6. Dragt, Alex J. (2005), "The symplectic group and classical mechanics", Annals of the New York Academy of Sciences, 1045 (1): 291–307, doi:10.1196/annals.1350.025, PMID 15980319. /wiki/Doi_(identifier) ↩

  7. Meyer, K. R.; Hall, G. R. (1991), Introduction to Hamiltonian dynamical systems and the N-body problem, Springer, ISBN 0-387-97637-X. 0-387-97637-X ↩

  8. Waterhouse, William C. (2005), "The structure of alternating-Hamiltonian matrices", Linear Algebra and its Applications, 396: 385–390, doi:10.1016/j.laa.2004.10.003. /wiki/William_C._Waterhouse ↩

  9. Ikramov, Khakim D. (2001), "Hamiltonian square roots of skew-Hamiltonian matrices revisited", Linear Algebra and its Applications, 325: 101–107, doi:10.1016/S0024-3795(00)00304-9. /wiki/Doi_(identifier) ↩

  10. Waterhouse, William C. (2005), "The structure of alternating-Hamiltonian matrices", Linear Algebra and its Applications, 396: 385–390, doi:10.1016/j.laa.2004.10.003. /wiki/William_C._Waterhouse ↩

  11. Paige, Chris; Van Loan, Charles (1981), "A Schur decomposition for Hamiltonian matrices", Linear Algebra and its Applications, 41: 11–32, doi:10.1016/0024-3795(81)90086-0. /wiki/Charles_F._Van_Loan ↩

  12. Waterhouse, William C. (2005), "The structure of alternating-Hamiltonian matrices", Linear Algebra and its Applications, 396: 385–390, doi:10.1016/j.laa.2004.10.003. /wiki/William_C._Waterhouse ↩