linear algebra Linear algebra is the branch of mathematics concerning linear equations such as: :a_1x_1+\cdots +a_nx_n=b, linear maps such as: :(x_1, \ldots, x_n) \mapsto a_1x_1+\cdots +a_nx_n, and their representations in vector spaces and through matrices ...

, skew-Hamiltonian matrices are special

matrices 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 ...

which correspond to skew-symmetric bilinear forms on a

symplectic vector space In mathematics, a symplectic vector space is a vector space ''V'' over a field ''F'' (for example the real numbers R) equipped with a symplectic bilinear form. A symplectic bilinear form is a mapping that is ; Bilinear: Linear in each argument ...

. Let ''V'' be a

vector space In mathematics and physics, a vector space (also called a linear space) is a set whose elements, often called '' vectors'', may be added together and multiplied ("scaled") by numbers called ''scalars''. Scalars are often real numbers, but can ...

, equipped with a

symplectic form In mathematics, a symplectic vector space is a vector space ''V'' over a field ''F'' (for example the real numbers R) equipped with a symplectic bilinear form. A symplectic bilinear form is a mapping that is ; Bilinear: Linear in each argument ...

\Omega

. Such a space must be even-dimensional. A linear map

A:\; V \mapsto V

is called a skew-Hamiltonian operator with respect to

\Omega

if the form

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

is skew-symmetric. Choose a basis

e_1, ... e_

in ''V'', such that

\Omega

is written as

\sum_i e_i \wedge e_

. Then a linear operator is skew-Hamiltonian with respect to

\Omega

if and only if its matrix ''A'' satisfies

A^T J = J A

, where ''J'' is the skew-symmetric matrix :

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

and ''I_n'' is the

n\times n

identity matrix. William C. Waterhouse
''The structure of alternating-Hamiltonian matrices''
Linear Algebra and its Applications, Volume 396, 1 February 2005, Pages 385-390 Such matrices are called skew-Hamiltonian. The square of a

Hamiltonian matrix In mathematics, a Hamiltonian matrix is a -by- matrix such that is symmetric, where is the skew-symmetric matrix :J = \begin 0_n & I_n \\ -I_n & 0_n \\ \end and is the -by- identity matrix. In other words, is Hamiltonian if and only if ...

is skew-Hamiltonian. The converse is also true: every skew-Hamiltonian matrix can be obtained as the square of a Hamiltonian matrix.

Heike Fassbender Heike Fassbender is a German mathematician specializing in numerical linear algebra. She is a professor in the Institute for Computational Mathematics at the Technical University of Braunschweig, and the president for the 2017–2019 term of the Ge ...

, D. Steven Mackey, Niloufer Mackey and Hongguo X
Hamiltonian Square Roots of Skew-Hamiltonian Matrices
Linear Algebra and its Applications 287, pp. 125 - 159, 1999

Notes

Matrices Linear algebra {{matrix-stub