In mathematics, a symplectic matrix is a matrix with real entries that satisfies the condition
(1)
where denotes the transpose of and is a fixed nonsingular, skew-symmetric matrix. This definition can be extended to matrices with entries in other fields, such as the complex numbers, finite fields, p-adic numbers, and function fields.
Typically is chosen to be the block matrix
where is the identity matrix. The matrix has determinant and its inverse is .
In mathematics, a symplecticmatrix is a 2 n × 2 n {\displaystyle 2n\times 2n} matrix M {\displaystyle M} with real entries that satisfies the condition...
this matrix being skew-symmetric, nonsingular, and hollow (all diagonal entries are zero). This should not be confused with a symplecticmatrix, which...
orthogonal group SO(n) Orthogonal matrix Semi-orthogonal matrix Quantum logic gate Special Unitary group SU(n) Symplecticmatrix Unitary group U(n) Unitary operator...
\omega } , called the symplectic form. The study of symplectic manifolds is called symplectic geometry or symplectic topology. Symplectic manifolds arise naturally...
the matrix exponential of a Hamiltonian matrix is symplectic. However the logarithm of a symplecticmatrix is not necessarily Hamiltonian because the...
n × n {\displaystyle n\times n} matrix with IID samples from the standard normal distribution. The Gaussian symplectic ensemble GSE ( n ) {\displaystyle...
{\displaystyle S\in {\text{Sp}}(2n)} is a symplecticmatrix and D is a nonnegative n-by-n diagonal matrix. Decomposition: A = B B {\displaystyle A=BB}...
In mathematics, a symplectic integrator (SI) is a numerical integration scheme for Hamiltonian systems. Symplectic integrators form the subclass of geometric...
Poisson matrix is defined as P ( ε ) = M J M T {\textstyle {\mathcal {P}}(\varepsilon )=MJM^{T}} , where J {\displaystyle J} is the symplecticmatrix under...
For the DARE, when A {\displaystyle A} is invertible, we define the symplecticmatrix Z = ( A + B R − 1 B T ( A − 1 ) T Q − B R − 1 B T ( A − 1 ) T − (...
matrices used in mathematics, science and engineering. A matrix (plural matrices, or less commonly matrixes) is a rectangular array of numbers called entries...
Lagrange matrix is defined as L ( η ) = M T J M {\textstyle {\mathcal {L}}(\eta )=M^{T}JM} , where J {\displaystyle J} is the symplecticmatrix under the...
matrix with unit determinant is a symplecticmatrix, and thus SL(2, Z) = Sp(2, Z), the symplectic group of 2 × 2 matrices. To find an explicit matrix...
In linear algebra, a symmetric matrix is a square matrix that is equal to its transpose. Formally, A is symmetric ⟺ A = A T . {\displaystyle A{\text{...
In mathematics and physics, a Hamiltonian vector field on a symplectic manifold is a vector field defined for any energy function or Hamiltonian. Named...
the matrix elements of U to real numbers [so that U is in the orthogonal group O(n)] or to real quaternion numbers [so that U is in the symplectic group...
to a factor of −1/2, involves the simplest (unimodular, symmetric) symplecticmatrix in Sp(2,R). That is, ( x , y ) M ( x y ) , {\displaystyle (x,y){\mathbf...
and a symplectic structure, which are required to be compatible (meaning that one uses the same J in the complex structure and the symplectic form, and...
the surface. This means that V is a 2g × 2g matrix with the property that V − VT is a symplecticmatrix. The Arf invariant of the knot is the residue...
unitary operator, then V admits an invariant complex symplectic form ω, and hence is a symplectic representation. This always holds if V is a representation...
equations gives the symplectic condition as: M J M T = J {\displaystyle MJM^{T}=J} The left hand side of the above is called the Poisson matrix of ε {\displaystyle...
mathematics and physics. Matrix groups or algebraic groups are (roughly) groups of matrices (for example, orthogonal and symplectic groups), and these give...
Matrix mechanics is a formulation of quantum mechanics created by Werner Heisenberg, Max Born, and Pascual Jordan in 1925. It was the first conceptually...
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