mathematics Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many ar ...

, a Hamiltonian matrix is a -by-

matrix Matrix (: matrices or matrixes) or MATRIX may refer to: Science and mathematics * Matrix (mathematics), a rectangular array of numbers, symbols or expressions * Matrix (logic), part of a formula in prenex normal form * Matrix (biology), the m ...

such that is

symmetric Symmetry () in everyday life refers to a sense of harmonious and beautiful proportion and balance. In mathematics, the term has a more precise definition and is usually used to refer to an object that is invariant under some transformations ...

, where is the

skew-symmetric matrix In mathematics, particularly in linear algebra, a skew-symmetric (or antisymmetric or antimetric) matrix is a square matrix whose transpose equals its negative. That is, it satisfies the condition In terms of the entries of the matrix, if a ...

J =
\begin
0_n    & I_n \\
-I_n & 0_n   \\
\end

and is the -by-

identity matrix In linear algebra, the identity matrix of size n is the n\times n square matrix with ones on the main diagonal and zeros elsewhere. It has unique properties, for example when the identity matrix represents a geometric transformation, the obje ...

. In other words, is Hamiltonian if and only if where denotes the

transpose In linear algebra, the transpose of a Matrix (mathematics), matrix is an operator which flips a matrix over its diagonal; that is, it switches the row and column indices of the matrix by producing another matrix, often denoted by (among other ...

.. (Not to be confused with

Hamiltonian (quantum mechanics) In quantum mechanics, the Hamiltonian of a system is an operator corresponding to the total energy of that system, including both kinetic energy and potential energy. Its spectrum, the system's ''energy spectrum'' or its set of ''energy eigenvalu ...

)

Properties

Suppose that the -by- matrix is written as the

block matrix In mathematics, a block matrix or a partitioned matrix is a matrix that is interpreted as having been broken into sections called blocks or submatrices. Intuitively, a matrix interpreted as a block matrix can be visualized as the original matrix w ...

A = \begin a & b \\ c & d \end

where , , , and are -by- matrices. Then the condition that be Hamiltonian is equivalent to requiring that the matrices and are symmetric, and that .. Another equivalent condition is that is of the form with symmetric. It follows easily from the definition that the transpose of a Hamiltonian matrix is Hamiltonian. Furthermore, the sum (and any

linear combination In mathematics, a linear combination or superposition is an Expression (mathematics), expression constructed from a Set (mathematics), set of terms by multiplying each term by a constant and adding the results (e.g. a linear combination of ''x'' a ...

) of two Hamiltonian matrices is again Hamiltonian, as is their

commutator In mathematics, the commutator gives an indication of the extent to which a certain binary operation fails to be commutative. There are different definitions used in group theory and ring theory. Group theory The commutator of two elements, ...

. It follows that the space of all Hamiltonian matrices is a

Lie algebra In mathematics, a Lie algebra (pronounced ) is a vector space \mathfrak g together with an operation called the Lie bracket, an alternating bilinear map \mathfrak g \times \mathfrak g \rightarrow \mathfrak g, that satisfies the Jacobi ident ...

, denoted . The dimension of is . The corresponding

Lie group In mathematics, a Lie group (pronounced ) is a group (mathematics), group that is also a differentiable manifold, such that group multiplication and taking inverses are both differentiable. A manifold is a space that locally resembles Eucli ...

is the

symplectic group In mathematics, the name symplectic group can refer to two different, but closely related, collections of mathematical groups, denoted and for positive integer ''n'' and field F (usually C or R). The latter is called the compact symplectic gr ...

. This group consists of the symplectic matrices, those matrices which satisfy . Thus, the

matrix exponential In mathematics, the matrix exponential is a matrix function on square matrix, square matrices analogous to the ordinary exponential function. It is used to solve systems of linear differential equations. In the theory of Lie groups, the matrix exp ...

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.. The

characteristic polynomial In linear algebra, the characteristic polynomial of a square matrix is a polynomial which is invariant under matrix similarity and has the eigenvalues as roots. It has the determinant and the trace of the matrix among its coefficients. The ...

of a real Hamiltonian matrix is even. Thus, if a Hamiltonian matrix has as an

eigenvalue In linear algebra, an eigenvector ( ) or characteristic vector is a vector that has its direction unchanged (or reversed) by a given linear transformation. More precisely, an eigenvector \mathbf v of a linear transformation T is scaled by a ...

, then , and are also eigenvalues. It follows that the trace of a Hamiltonian matrix is zero. The square of a Hamiltonian matrix is skew-Hamiltonian (a matrix is skew-Hamiltonian if ). Conversely, every skew-Hamiltonian matrix arises as the square of a Hamiltonian matrix..

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 is Hamiltonian if , as above. Another possibility is to use the condition where the superscript asterisk () denotes the

conjugate transpose In mathematics, the conjugate transpose, also known as the Hermitian transpose, of an m \times n complex matrix \mathbf is an n \times m matrix obtained by transposing \mathbf and applying complex conjugation to each entry (the complex conjugate ...

Hamiltonian operators

Let be a vector space, equipped with a symplectic form . A linear map

A : \; V \mapsto V

is called a Hamiltonian operator with respect to if the form

x, y \mapsto \Omega(A(x), y)

is symmetric. Equivalently, it should satisfy :

\Omega(A(x), y) = -\Omega(x, A(y))

Choose a basis in , such that is written as

\sum_i e_i \wedge e_

. A linear operator is Hamiltonian with respect to if and only if its matrix in this basis is Hamiltonian.

References

{{Matrix classes Matrices (mathematics)