In
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'' (franchis ...
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, if a_ ...
:
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 o ...
. 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 ...
:
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, a ...
. 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 Binary operation, operation called the Lie bracket, an Alternating multilinear map, alternating bilinear map \mathfrak g \times \mathfrak g \rightarrow ...
, 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 additio ...
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 grou ...
. 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 gives ...
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 chara ...
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 w ...
. 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 denoted b ...
, then , and are also eigenvalues.
It follows that the
trace
Trace may refer to:
Arts and entertainment Music
* Trace (Son Volt album), ''Trace'' (Son Volt album), 1995
* Trace (Died Pretty album), ''Trace'' (Died Pretty album), 1993
* Trace (band), a Dutch progressive rock band
* The Trace (album), ''The ...
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 con ...
.
[.]
Hamiltonian operators
Let be a vector space, equipped with a symplectic form . A linear map
is called a Hamiltonian operator with respect to if the form
is symmetric. Equivalently, it should satisfy
:
Choose a basis in , such that is written as
. A linear operator is Hamiltonian with respect to if and only if its matrix in this basis is Hamiltonian.
References
{{Matrix classes
Matrices