mathematics Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...

, a Hamiltonian matrix is a -by-

matrix Matrix most commonly refers to: * ''The Matrix'' (franchise), an American media franchise ** '' The Matrix'', a 1999 science-fiction action film ** "The Matrix", a fictional setting, a virtual reality environment, within ''The Matrix'' (franchi ...

such that is

symmetric Symmetry (from grc, συμμετρία "agreement in dimensions, due proportion, arrangement") in everyday language refers to a sense of harmonious and beautiful proportion and balance. In mathematics, "symmetry" has a more precise definiti ...

, 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, i ...

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. Terminology and notation The identity matrix is often denoted by I_n, or simply by I if the size is immaterial or ...

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

transpose In linear algebra, the transpose of a 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 notations). The tr ...

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 mat ...

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) 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 identi ...

, denoted . The dimension of is . The corresponding

Lie group In mathematics, a Lie group (pronounced ) is a group that is also a differentiable manifold. A manifold is a space that locally resembles Euclidean space, whereas groups define the abstract concept of a binary operation along with the addi ...

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 gro ...

. 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 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 exponential give ...

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 c ...

of a real Hamiltonian matrix is

even Even may refer to: General * Even (given name), a Norwegian male personal name * Even (surname) * Even (people), an ethnic group from Siberia and Russian Far East **Even language, a language spoken by the Evens * Odd and Even, a solitaire game wh ...

. Thus, if a Hamiltonian matrix has as an

eigenvalue In linear algebra, an eigenvector () or characteristic vector of a linear transformation is a nonzero vector that changes at most by a scalar factor when that linear transformation is applied to it. The corresponding eigenvalue, often denote ...

, 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

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 \boldsymbol is an n \times m matrix obtained by transposing \boldsymbol and applying complex conjugate on each entry (the complex c ...

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