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If the characteristic of the field is 2, then a skew-symmetric matrix is the same thing as a symmetric matrix. The sum of two skew-symmetric matrices is skew-symmetric. A scalar multiple of a skew-symmetric matrix is skew-symmetric. The elements on the diagonal of a skew-symmetric matrix are zero, and therefore its trace equals zero.
Similarly in characteristic different from 2, each diagonal element of a skew-symmetric matrix must be zero, since each is its own negative. In linear algebra, a real symmetric matrix represents a self-adjoint operator [1] represented in an orthonormal basis over a real inner product space.
However, any rotation (special orthogonal) matrix Q can be written as = (() (+)) for some skew-symmetric matrix A; more generally any orthogonal matrix Q can be written as = (+) for some skew-symmetric matrix A and some diagonal matrix E with ±1 as entries. [4] A slightly different form is also seen, [5] [6] requiring different mappings in ...
Skew-Hermitian matrix: A square matrix which is equal to the negative of its conjugate transpose, A * = −A. Skew-symmetric matrix: A matrix which is equal to the negative of its transpose, A T = −A. Skyline matrix: A rearrangement of the entries of a banded matrix which requires less space. Sparse matrix: A matrix with relatively few non ...
In linear algebra, a skew-Hamiltonian matrix is a specific type of matrix that corresponds to a skew-symmetric bilinear form on a symplectic vector space. Let be a vector space equipped with a symplectic form, denoted by Ω. A symplectic vector space must necessarily be of even dimension.
Then the condition that A be Hamiltonian is equivalent to requiring that the matrices b and c are symmetric, and that a + d T = 0. [1] [2] Another equivalent condition is that A is of the form A = JS with S symmetric. [2]: 34 It follows easily from the definition that the transpose of a Hamiltonian matrix is Hamiltonian.
The difference of a square matrix and its conjugate transpose () is skew-Hermitian (also called antihermitian). This implies that the commutator of two Hermitian matrices is skew-Hermitian. An arbitrary square matrix C can be written as the sum of a Hermitian matrix A and a skew-Hermitian matrix B.
Upon fixing a basis for V, the symplectic group becomes the group of 2n × 2n symplectic matrices, with entries in F, under the operation of matrix multiplication. This group is denoted either Sp(2n, F) or Sp(n, F). If the bilinear form is represented by the nonsingular skew-symmetric matrix Ω, then